G. I. Taylor was a happy man who spent a long life doing what he wanted most to do and doing it supremely well. He was a natural scientist whose character and activities were perfectly matched, and that allowed the fullest use of his creative talents. This short article is intended to help the understanding of G. I. Taylor's approach to his research by providing some information about the man and his life, in particular the major events in his early adult years which determined the direction of much of his subsequent research. A list of the now rather numerous articles about the man and his family and his work is appended.

Several aspects of the use of digital computers to generate solutions of equations of interest to fluid mechanics are discussed. The inter-disciplinary nature of the field of computational fluid dynamics (CFD) is emphasized: the dependence on strides in computer technology, the impact of advances in algorithm development, the continuous interaction with laboratory experiment and analytical theory. The particular role of that mode of computer usage usually referred to as the numerical experiment is highlighted. ‘Experiments’ of this type have played a central role in establishing concepts such as the soliton and the strange attractor as paradigms within fluid mechanics. The ambitious goal of providing digital counterparts to laboratory equipment such as the wind tunnel is considered. The possibility of abandoning the Eulerian representation of flow fields in favour of following swarms of Lagrangian particles on a computer is stressed. Issues arising from and results of using this methodology are reviewed. Computer simulations are contrasted with computer generated animation. The paper concludes with speculations on future developments.

There has been an increased interest in ocean phenomena with horizontal scales comparable to the radius of the Earth, and timescales of years and beyond. These phenomena occur in the presence of intense processes of higher spatial and temporal frequency. An observational programme for the large-scale phenomena has an inherent advantage if it can rely on measurements that are, by their very nature, integrated moments over the prerequisite scale.

The oceans provide an excellent medium for transmitting sound waves of low frequency, as demonstrated in the closing days of World War II, and subsequently confirmed by a 20000 km acoustic transmission between Perth, Australia and Bermuda. For the last six years we have been developing a method (Ocean Acoustic Tomography) to take advantage of the favourable ocean acoustic properties. We measure travel time Δ+ from mooring m to mooring n (positive x), and Δ− from n to m. The sum Δ+ + Δ− then gives information about the sound speed C (e.g. temperature) averaged along the acoustic ray path; the difference Δ+ – Δ− gives information about the x-component u of current velocity. The recorded acoustic signal can be decomposed into 10-20 distinct ray arrivals Δi, each with a distinct ray path and associated depth-weighting of the ocean column; the ray travel times can be inverted to yield information about the depth profiles C(z) and u(z). The product 〈C〉 〈u〉 of these range-averaged quantities is related to the climatological large-scale heat flux; the space-time average $\overline{\langle \delta C\,\delta u\rangle}$ is related to the eddy heat flux, and can be estimated by measuring the difference variance (Δ+) – variance (Δ−).

In this paper the authors present a numerical and experimental analysis of the winged keel originally developed for the International twelve-metre class yacht Australia II that won the America's Cup in 1983. After briefly explaining why this keel was evolved in 1981, some basic considerations are presented relating keel performance to various design parameters. The results of numerical flow analyses and wind-tunnel measurements on a model of a winged keel are then presented and compared. The differences between the performance with and without winglets fitted to the keel are discussed. The fitting of winglets appreciably enhances the performance of a low-aspect-ratio lifting surface such as the keel of a twelve-metre yacht.

The phenomenon of interfacial motion between two immiscible viscous fluids in the narrow gap between two parallel plates (Hele-Shaw cell) is considered. This flow is currently of interest because of its relation to pattern selection mechanisms and the formation of fractal, structures in a number of physical applications. Attention is concentrated on the fingers that result from the instability when a less-viscous fluid drives a more-viscous one. The status of the problem is reviewed and progress with the thirty-year-old problem of explaining the shape and stability of the fingers is described. The paradoxes and controversies are both mathematical and physical. Theoretical results on the structure and stability of steady shapes are presented for a particular formulation of the boundary conditions at the interface and compared with the experimental phenomenon. Alternative boundary conditions and future approaches are discussed.

We consider the motion of the flattened bubbles which form when air is injected into a viscous fluid contained in the narrow gap between two flat, parallel plates which make up a conventional Hele-Shaw cell, inclined at an angle x to the horizontal. We present a number of qualitative observations on the formation and interaction of the streams of bubbles that appear when air is injected continuously into the cell. The majority of this paper is then concerned with the shape and velocity of rise of single, isolated bubbles over a wide range of bubble size and cell inclination. We compare these results to theories by Taylor & Saffman (1959), and Tanveer (1986). It appears that the bubble characteristics found by an ad hoc speculation in Taylor & Saffman (1959) and by Tanveer (1986) only agree with the experimental results in the limit α → 0, and for large bubble widths (D). For finite values of α, it is necessary to use the measured bubble shape in order to calculate the rise velocity using the more general Taylor & Saffman (1959) formulation. Deviations from these theories for small D can be explained by considering the effects of the detailed flow close to the bubble surface.

An energy stability theory is formulated for systems having moving contact lines. The method derives from criteria obtained from the integral mechanical-energy balance manipulated to reflect general material and dynamical properties of moving-contact-line regions. The method yields conditions for both stability and instability and is applied to the two-dimensional Rayleigh-Taylor problem in a vertical slot.

A computer-controlled four-roll mill is used to examine two transient modes of deformation of a liquid drop: elongation in a steady flow and interfacial-tension-driven motion which occurs after the flow is stopped abruptly. For modest extensions, drop breakup does not occur with the flow on, but may occur following cessation of the flow as a result of deterministic motions associated with internal pressure gradients established by capillary forces. These relaxation and breakup phenomena depend on the initial drop shape and the relative viscosities of the two fluids. Capillary-wave instabilities at the fluid-fluid interface are observed only for highly elongated drops. This study is a natural extension of G. I. Taylor's original studies of the deformation and burst of droplets in well-defined flow fields.

This paper is concerned with the transient motion produced when a viscous incompressible fluid is forced from an initial state of rest. The applied force is time dependent in the form of an impulse, step and ramp function applied at a point and along a line. These cases have been chosen because they form a logical progression for investigating the connection between the flow Reynolds number and the sequence of events leading to the creation of a starting vortex. Much of the structure of the starting process can be revealed through a study of boundary conditions, integrals of the motion and the invariance properties of the governing equations prior to the consideration of a particular solution. The method used to bring out the flow structure is applicable to flows that can be treated as self-similar over some interval in time. The equations for unsteady particle paths are written in terms of similarity variables and then analysed as a quasi-autonomous system with the, usually time-dependent, Reynolds number treated as a parameter. The structure of the flow is examined by finding and classifying critical points in the phase portrait of this system. Bifurcations in the phase portrait are found to occur at specific values of the Reynolds number of the flow in question. When exact solutions of the Stokes equations for the low-Reynolds-number limit are examined they are found to contain two critical Reynolds numbers and three distinct states of motion which culminate in the onset of a vortex roll-up. An interesting feature of the Stokes solutions for planar unsteady jets is that they are uniformly valid over 0 < r < ∞.

Gravitational and viscous torques acting on swimming micro-organisms orient their trajectories. The horizontal component of the swimming velocity of individuals of the many algal genera having a centre of mass displaced toward the rear of the cell is therefore in the direction g × ([dtri ] × u), where g is the acceleration due to gravity. This phenomenon, called gyrotaxis, results in the cells swimming toward downward-flowing regions of their environment. Since the cells’ density is greater than that of water, regions of high (low) cell concentration sink (rise). The horizontal component of gyrotaxis reinforces this type of buoyant convection, whilst the vertical one maintains it. Gyrotactic buoyant convection results in the spontaneous generation of descending plumes containing high cell concentration, in spatially regular concentration/convection patterns, and in the perturbation of initially well-defined flow fields. This paper presents a height- and azimuth-independent steady-state solution of the Navier-Stokes and cell conservation equations. This solution, and the growth rate of a concentration fluctuation, are shown to be governed by a parameter similar to a Rayleigh number.

An algorithm has been developed which enables local Taylor-series-expansion solutions of the Navier-Stokes and continuity equations to be generated to arbitrary order. Much of the necessary algebra for generating these solutions can be done on a computer. Various properties of the algorithm are investigated and checked by making comparisons with known solutions of the equations of motion. A method of synthesizing nonlinear viscous-flow patterns with certain required properties is developed and applied to the construction of a number of two- and three-dimensional flow-separation patterns. These patterns are asymptotically exact solutions of the equations of motion close to the origin of the expansion. The region where the truncated series solution satisfies the full equations of motion to within a specified accuracy can be found.

Two-dimensional turbulence is investigated experimentally in thin liquid films. This study shows the spontaneous formation of couples of opposite-sign vortices in von Kármán wakes. The structure of these couples, their behaviour and their role in turbulent flows is then studied using both a numerical simulation and laboratory results.

The response of a convective flow to spatially periodic forcing at a period different from the critical wavelength is investigated experimentally. For reasons of experimental convenience, we utilize an electrohydrodynamic instability in a thin layer of nematic liquid. With this system, a sample containing several hundred rolls is easily obtained, and periodic forcing is imposed electrically using a specially designed interdigitated electrode. Several novel flows with multiple periodicities are found. They may be broadly classified as commensurate (phase-locked) or incommensurate (quasi-periodic) flows, depending on whether the dominant periodicity of the perturbed flow and the periodicity of the forcing are in the ratio of small integers. Near the instability threshold and for weak forcing, an approximation of slow spatial variations is satisfied. In that case, the flows can be described quantitatively by an amplitude equation. We note a close connection between these hydrodynamic phenomena and a problem of competing periodicities that occurs in statistical mechanics. This relationship leads us to suggest the use of a discontinuous function for describing certain incommensurate flows in which non-repeating short and long groups of rolls are observed to occur in an irregular sequence. We also note, as predicted by Pal & Kelly, that two-dimensional forcing can lead to a three-dimensional flow.

Experiments of Andereck et al. (1986) with corotating cylinders, show that Taylor-vortex flow (TVF) can bifurcate into one of the following cellular flows: wavy vortices (WV), twisted vortices (TW), wavy inflow boundaries (WIB), wavy outflow boundaries (WOB). We describe here the structure of these different flows, showing how they result from simple symmetry breaking. Moreover we consider the codimension-two situation where WIB and WOB interact, since this is an observed physical situation.

The method used in this paper is based on symmetry arguments. It differs notably from the Liapunov-Schmidt reduction used in particular by Golubitsky & Stewart (1986) on the same problem with counter-rotating cylinders. Here we take into account all the dynamics, instead of restricting the study to oscillating solutions. In addition to the standard oscillatory modes, we have a translational mode due to the indeterminacy of TVF under the shifts along the axis. We derive an amplitude-expansion procedure which allows the translational mode to depend on time. Our amplitude equations have nevertheless a simple structure because the oscillatory modes have a precise symmetry. They break, in general, the rotational invariance and they are either symmetric or antisymmetric with respect to the plane z = 0. Moreover, the most typical cases are when either of these modes has the same axial period as TVF or when their axial period is double this. This leads to four different cases which are shown to give WV, TW, WIB or WOB, all these flows being ‘rotating waves’, i.e. they are steady in a suitable rotating frame.

Finally we consider the interaction between WIB and WOB that occurs when, at the onset of instability, the two critical modes arise simultaneously. In this case we show in particular that there may exist a stable quasi-periodic flow bifurcating from WIB or WOB. The two main frequencies are those of underlying WIB and WOB, while there may exist a third frequency corresponding to a slow superposed travelling wave in the axial direction.

The method was used in the counter-rotating case for interacting non-axisymmetric modes (see Chossat et al. 1986). One of the original contributions here is not only to clarify the origin of all observed bifurcations from TVF, but also to handle the translational mode which may not stay small. This technique combined with centre-manifold and equivariance techniques may be helpful for many problems starting with orbits of solutions, such as the TVF considered here.

A wide class of solutions of the steady Euler equations, representing localized rotational disturbances imbedded in a uniform stream U0 is inferred by considering the process of magnetic relaxation to analogous magnetostatic equilibria. These solutions, which may be regarded as generalizations of vortex rings, are characterized by their streamline topology, distinct topologies giving rise to distinct solutions.

Particular attention is paid to the class of axisymmetric solutions described by Stokes stream function ψ(s, z). It is argued that the appropriate topological ‘invariant’ characterizing the flow is the function Vψ representing the volume inside toroidal surfaces ψ = const, in the region of closed streamlines where ψ > 0. This function is described as the ‘signature’ of the flow, and it is shown that in a certain sense, flows with different signatures are topologically distinct. The approach yields a method by which flows of arbitrary signature V(ψ) may in principle be found, and the corresponding vorticity ωφ = sFψ calculated.

This is a personal statement on the present state of understanding of coherent structures, in particular their spatial details and dynamical significance. The characteristic measures of coherent structures are discussed, emphasizing coherent vorticity as the crucial property. We present here a general scheme for educing structures in any transitional or fully turbulent flow. From smoothed vorticity maps in convenient flow planes, this scheme recognizes patterns of the same mode and parameter size, and then phase-aligns and ensemble-averages them to obtain coherent structure measures. The departure of individual realizations from the ensemble average denotes incoherent turbulence. This robust scheme has been used to educe structures from velocity data using a rake of hot wires as well as direct numerical simulations and can educe structures using newer measurement techniques such as digital image processing. Our recent studies of coherent structures in several free shear flows are briefly reviewed. Detailed data in circular and elliptic jets, mixing layers, and a plane wake reveal that incoherent turbulence is produced at the ‘saddles’ and then advected to the ‘centres’ of the structures. The mechanism of production of turbulence in shear layers is the stretching of longitudinal vortices or ‘ribs’ which connect the predominantly spanwise ‘rolls’; the ribs induce spanwise contortions of rolls and cause mixing and dissipation, mostly at points where they connect with rolls. We also briefly discuss the role of coherent structures in aerodynamic noise generation and argue that the structure breakdown process, rather than vortex pairing, is the dominant mechanism of noise generation. The ‘cut-and-connect’ interaction of coherent structures is proposed as a specific mechanism of aerodynamic noise generation, and a simple analytical model of it shows that it can provide acceptable predictions of jet noise. The coherent-structures approach to turbulence, apart from explaining flow physics, has also enabled turbulence management via control of structure evolution and interactions. We also discuss some new ideas under investigation: in particular, helicity as a characteristic property of coherent structures.

Speculations abound that several facets of fully developed turbulent flows are fractals. Although the earlier leading work of Mandelbrot (1974, 1975) suggests that these speculations, initiated largely by himself, are plausible, no effort has yet been made to put them on firmer ground by, resorting to actual measurements in turbulent shear flows. This work is an attempt at filling this gap. In particular, we examine the following questions: (a) Is the turbulent/non-turbulent interface a self-similar fractal, and (if so) what is its fractal dimension ? Does this quantity differ from one class of flows to another? (b) Are constant-property surfaces (such as the iso-velocity and iso-concentration surfaces) in fully developed flows fractals? What are their fractal dimensions? (c) Do dissipative structures in fully developed turbulence form a fractal set? What is the fractal dimension of this set? Answers to these questions (and others to be less fully discussed here) are interesting because they bring the theory of fractals closer to application to turbulence and shed new light on some classical problems in turbulence - for example, the growth of material lines in a turbulent environment. The other feature of this work is that it tries to quantify the seemingly complicated geometric aspects of turbulent flows, a feature that has not received its proper share of attention. The overwhelming conclusion of this work is that several aspects of turbulence can be described roughly by fractals, and that their fractal dimensions can be measured. However, it is not clear how (or whether), given the dimensions for several of its facets, one can solve (up to a useful accuracy) the inverse problem of reconstructing the original set (that is, the turbulent flow itself).

We present a review of quantum turbulence, that is, the turbulent motion of quantized vortex lines in superfluid helium. Our discussion concentrates on the turbulence produced by steady, uniform heat flow in a pipe, but touches on other turbulent flows as well. We have attempted to motivate the study of quantum turbulence and discuss briefly its connection with classical turbulence. We include background on the two-fluid model and mutual friction theory, examples of modern experimental techniques, and a brief survey of the phenomenology. We discuss the important recent insights that vortex dynamics has provided to the understanding of quantum turbulence, from simple scaling arguments to detailed numerical simulations. We conclude with a discussion of open questions in this field.

The entrainment assumption, relating the inflow velocity to the local mean velocity of a turbulent flow, has been used successfully to describe natural phenomena over a wide range of scales. Its first application was to plumes rising in stably stratified surroundings, and it has been extended to inclined plumes (gravity currents) and related problems by adding the effect of buoyancy forces, which inhibit mixing across a density interface. More recently, the influence of viscosity differences between a turbulent flow and its surroundings has been studied. This paper surveys the background theory and the laboratory experiments that have been used to understand and quantify each of these phenomena, and discusses their applications in the atmosphere, the ocean and various geological contexts.

An experiment is described in which a turbulent boundary layer was generated along a sloping wall of a laboratory tank containing salt-stratified fluid. Initially, the pycnocline separating the upper fresh-water layer from the lower saline layer was relatively thin; it thickened in response to the mixing as the experiment proceeded. Two types of mean circulation developed as a result of the boundary mixing. In the boundary layer, counterflowing mean streams were observed that augmented the diffusion of salt up- and downslope. This augmented dispersion in turn tended to spread the salt beyond the depth range of the pycnocline in the ambient fluid, producing a mean convergence in the boundary layer and intrusions from the layer into the ambient fluid.

The evolving density structure in the ambient fluid was measured by conductivity-probe traverses, from which the net buoyancy flux (or salt flux) and volume flux in the boundary layer were determined. The level of zero volume flux was found to coincide closely with the level of maximum stability frequency N, so that the buoyancy transport across this level was entirely the result of turbulent dispersion in the boundary layer. At other levels, the convective transports up and down contribute significantly. A simple theory provides scaling in terms of the laboratory parameters; in terms of the inferred overall turbulent viscosity νe, the buoyancy transport resulting from boundary-layer dispersion was found to be \[ F_{\rm B} = 0.60 \left\{\frac{\nu_{\rm e}N(z)}{\sin\theta}\right\}^{\frac{3}{2}}\cos^2\theta, \] and the intrusion velocity into the ambient fluid is \[ v_1 = 0.42\frac{\nu_{\rm e}^{\frac{3}{2}}}{N^{\frac{1}{2}}h^2}\frac{\cos^2\theta}{(\sin\theta)^{\frac{3}{2}}}, \] where θ is the angle of the sloping bed. These expressions must become invalid when θ becomes vanishingly small; their application to estuarine and continental slope flows is discussed briefly.