This and the subsequent part of the Journal of Fluid Mechanics will be devoted to the publication of papers on turbulence. Each of these papers was the subject of an oral report at a Symposium on Turbulence sponsored by the Boeing Company and organized by the Flight Sciences Laboratory of the Boeing Scientific Research Laboratories, a t Seattle, Washington, U.S.A., from 23 to 27 June 1969. The Organizing Committee consisted of Dr A. Goldburg, Prof. L. S. G. Kovasznay (Chairman), andDr Y.-H. Pao (Secretary). At this Symposium invited participants listened both to review lectures and to contributed papers, and tried by discussion to assess the present position in the study of turbulence.

This paper considers the stability of liquid metal drops subject to a high-frequency AC magnetic field. An energy variation principle is derived in terms of the surface integral of the scalar magnetic potential. This principle is applied to a thin perfectly conducting liquid disk, which is used to model the drops constrained in a horizontal gap between two parallel insulating plates. Firstly, the stability of a circular disk is analysed with respect to small-amplitude harmonic edge perturbations. Analytical solution shows that the edge deformations with the azimuthal wavenumbers m = 2, 3, 4, . . . start to develop as the magnetic Bond number exceeds the critical threshold Bmc = 3π(m + 1)/2. The most unstable is m = 2 mode, which corresponds to an elliptical deformation. Secondly, strongly deformed equilibrium shapes are modelled numerically by minimising the associated energy in combination with the solution of a surface integral equation for the scalar magnetic potential on an unstructured triangular mesh. The edge instability is found to result in the equilibrium shapes of either two- or threefold rotational symmetry depending on the magnetic field strength and the initial perturbation. The shapes of higher rotational symmetries are unstable and fall back to one of these two basic states. The developed method is both efficient and accurate enough for modelling of strongly deformed drop shapes.

In this paper we study both theoretically and experimentally the inverse problem of indirectly measuring the shape of a localized bottom deformation with a non-instantaneous time evolution, from either an instantaneous global state (space-based inversion) or a local time-history record (time-based inversion) of the free-surface evolution. Firstly, the mathematical inversion problem is explicitly defined and uniqueness of its solution is established. We then show that this problem is ill-posed in the sense of Hadamard, rendering its solution unstable. In order to overcome this difficulty, we introduce a regularization scheme as well as a strategy for choosing the optimal value of the associated regularization parameter. We then conduct a series of laboratory experiments in which an axisymmetric three-dimensional bottom deformation of controlled shape and time evolution is imposed on a layer of water of constant depth, initially at rest. The detailed evolution of the air–liquid interface is measured by means of a free-surface profilometry technique providing space- and time-resolved data. Based on these experimental data and employing our regularization scheme, we are able to show that it is indeed possible to reconstruct the seabed profile responsible for the linear free-surface dynamics either by space- or time-based inversions. Furthermore, we discuss the different relative advantages of each type of reconstruction, their associated errors and the limitations of the inverse determination.

Observations in experiments and simulations show that the kinematic behaviour of an elastic capsule, suspended and rotating in shear flow, depends upon the flow strength, the capsule membrane material properties and its at-rest shape. We develop a linear stability description of the periodically rotating base state of this coupled system, as represented by a boundary integral flow formulation with spherical harmonic basis functions describing the elastic capsule geometry. This yields Floquet multipliers that classify the stability of the capsule motion for elastic capillary numbers $Ca$ ranging from $Ca=0.01$ to 5. Viscous dissipation rapidly damps most perturbations. However, for all cases, a single component grows or decays slowly, depending upon $Ca$, over many periods of the rotation. The transitions in this stability behaviour correspond to the different classes of rotating motion observed in previous studies.

We develop a bifurcation theory describing the conditions under which vortices are created or destroyed in a two-dimensional incompressible flow. We define vortices using the $Q$-criterion and analyse the vortex structure by considering the evolution of the zero contours of $Q$. The theory identifies topological changes of the vortex structure and classifies these as four possible types of bifurcations, two occurring away from boundaries, and two occurring near no-slip walls. Our theory provides a description of all possible codimension-one bifurcations where time is treated as the bifurcation parameter. To illustrate our results, we consider the early stages of boundary layer eruption at moderate Reynolds numbers in the range from $Re=750$ to $Re=2250$. By analysing numerical simulations of the phenomenon, we show how to describe the eruption process as sequences of the four possible bifurcations of codimension one. Our simulations show that there is a single codimension-two point within our parameter range. This codimension-two point arises at $Re=1817$ via the coalescence of two codimension-one bifurcations which are associated with the creation and subsequent destruction of one of the vortices that erupt from the boundary layer. We present a theoretical description of this process and explain how the occurrence of this phenomenon separates the parameter space into two regions with distinct evolution of the topology of the vortices.

Steady incompressible flow down a slowly curving circular pipe is considered, analytically and numerically. Both real and complex solutions are investigated. Using high-order Hermite–Padé approximants, the Dean series solution is analytically continued outside its circle of convergence, where it predicts a complex solution branch for real positive Dean number, $K$. This is confirmed by numerical solution. It is shown that other previously unknown solution branches exist for all $K>0$, which are related to an unforced complex eigensolution. This non-uniqueness is believed to be generic to the Navier–Stokes equations in most geometries. By means of path continuation, numerical solutions are followed around the complex $K$-plane. The standard Dean two-vortex solution is shown to lie on the same hypersurface as the eigensolution and the four-vortex solutions found in the literature. Elliptic pipes are considered and shown to exhibit similar behaviour to the circular case. There is an imaginary singularity limiting convergence of the Dean series, an unforced solution at $K=0$ and non-uniqueness for $K>0$, culminating in a real bifurcation.

George Batchelor was one of the giants of fluid mechanics in the second half of the twentieth century. He had a passion for physical and quantitative understanding of fluid flows and a single-minded determination that fluid mechanics should be pursued as a subject in its own right. He once wrote that he ‘spent a lifetime happily within its boundaries’. Six feet tall, thin and youthful in appearance, George's unchanging attire and demeanour contrasted with his ever-evolving scientific insights and contributions. His strongly held and carefully articulated opinions, coupled with his forthright objectivity, shone through everything he undertook.

George's pervasive influence sprang from a number of factors. First, he conducted imaginative, ground-breaking research, which was always based on clear physical thinking. Second, he founded a school of fluid mechanics, inspired by his mentor G. I. Taylor, that became part of the world renowned Department of Applied Mathematics and Theoretical Physics (DAMTP) of which he was the Head from its inception in 1959 until he retired from his Professorship in 1983. Third, he established this Journal in 1956 and actively oversaw all its activities for more than forty years, until he relinquished his editorship at the end of 1998. Fourth, he wrote the monumental textbook An Introduction to Fluid Dynamics, which first appeared in 1967, has been translated into four languages and has been relaunched this year, the year of his death. This book, which describes the fundamentals of the subject and discusses many applications, has been closely studied and frequently cited by generations of students and research workers. It has already sold over 45 000 copies. And fifth, but not finally, he helped initiate a number of international organizations (often European), such as the European Mechanics Committee (now Society) and the biennial Polish Fluid Mechanics Meetings, and contributed extensively to the running of IUTAM, the International Union of Theoretical and Applied Mechanics. The aim of all of these associations is to foster fluid (and to some extent solid) mechanics and to encourage the development of the subject.

Nonlinear effects on the receptivity of cross-flow in the swept Hiemenz boundary layer are investigated. Numerical simulations are generated using a vorticity form of the Navier–Stokes equations. Steady perturbations are established using surface suction and blowing distributed along the spanwise direction as either a periodic strip or a band of small holes. The method of excitation, the size and the location of the prescribed forcing are shown to have a significant influence on the receptivity of the boundary layer. Blowing holes are found to excite perturbations with considerably larger magnitudes than those generated using a periodic suction and blowing strip. A semi-logarithmic relationship is derived that relates the initial amplitude of the linear-only disturbances with the location at which the absolute magnitude of the chordwise primary Fourier harmonic attains a stationary point or a size of approximately one-tenth of the free-stream spanwise velocity. Furthermore, the size of the physical chordwise velocity perturbation about this position can be estimated directly from the linear-only solutions. This would suggest that, for sufficiently small initial amplitudes, the onset of some nonlinear flow development properties can be predicted directly from a linear receptivity analysis.

Recent experimental observation of supercooling in large hypersonic wind tunnels using pure nitrogen identified a broad range of non-equilibrium metastable vapour states of the flow in the test cell. To investigate this phenomenon a number of real-gas effects are analysed and compared with predictions made using the ideal-gas equation of state and equilibrium thermodynamics. The observed limit on the extent of supercooling is found to be at 60% of the temperature difference from the sublimation line to Gibbs’ absolute limit on phase stability. The mass fraction then condensing is calculated to be 12–14%. Included in the study are virial effects, quantization of rotational and vibrational energy, and the possible role of vibrational relaxation and freezing in supercooling. Results suggest that use of the supercooled region to enlarge the Mach–Reynolds number test envelope may be practical. Data from model tests in supercooled flows support this possibility.

The separation points which occur in the Newtonian theory of hypersonic flow are treated by locally modifying the shock-layer equations. This approach leads to direct verification of the free-layer theory for separation on certain bodies with discontinuous curvature. Separation points on bodies with continuous curvature have already been treated by matching the upstream shock-layer solution to the downstream free-layer solution. The agreement that is found between these results and those of the present approach provides further confirmation of the free-layer theory.

This paper is concerned with understanding the development of a free surface and the velocity of a thick creeping film emerging from a slit onto a plate. There is a difference between free surface flows with film thickness of 10 to 20 mm and thin film flows of less than 1 to 2 mm due to surface tension influence.

A laser–Doppler velocimeter of high spatial and temporal resolution has been used to measure the decreasing thickness of slowly flowing liquid and the velocity profiles at the exit of the slit and in the transition region of the plate. The hydrodynamic entrance length can be calculated from an empirical dimensionless relation that includes only the plate inclination angle β as a parameter.

The investigation by Hayes (1964a) of the behaviour of a constant-density inviscid rotational flow in the neighbourhood of a stagnation point on a plane wall has been extended to include the effects of viscosity. The principal effect is the manner in which the singularity in vorticity discovered by Hayes is removed. A solution of only the boundary-layer equatiosn indicates the vorticity decays algebraically from the wall. Application of the method of matched asymptotic expansions, however, shows that the difference between boundary layer and outer vorticity, when carried out to second order in the outer flow, does not contribute to an algebraic decay. These results suggest that an infinite number of higher-order outer terms are generated which match the algebraic terms thereby yielding the conventional exponential decay. Numerical results are presented which also support this conclusion. The main contribution of the wall shear stress in the immediate neighbourhood of the stagnation point is shown to come from the external lateral vorticity.

A generalized linear theory is developed for a compressible shear flow by extending the Fourier integral representations for the pressure field disturbed by pressure dipoles, and it is applied to a transonic shear flow past a thin aerofoil.

A method of determining the pressure distribution on the aerofoil surface is shown. Some numerical examples are presented and discussed.

It is found that the three-dimensional effect due to non-uniformity of the Mach number appears in the greatest degree at lower supersonic span-stations. Even within the scope of the linearized theory not such a great lift force arises near the sonic station as would be expected from the linearized uniform flow theory.

The conditions under which stationary waves may exist on a stream of water of finite depth are investigated theoretically for the case of a current which is uniform except for a constant defect in velocity in a region near the free surface. The analysis is extended to provide a two-dimensional theory for the surface profile induced by a simplified model of a hovering craft. The relevance of this work to the use of high speed flumes is discussed, and an example demonstrates the importance of the velocity distribution near the free surface.