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.

The generation of ship bow waves is studied within the framework of potential flow theory. Assuming the ship bow to be slender, or thin, a pattern of the flow is derived using the method of matched asymptotic expansions. This method leads to the determination of three different zones in which three asymptotic expansions are performed and matched. To first order with respect to the slenderness parameter, the near-field flow appears to be two-dimensional in each transverse plane along the bow. However, it is demonstrated that three-dimensional effects are important in front of the ship and must be taken into account in the composite solution. This leads to a three-dimensional correction to be added to the two-dimensional solution along the ship. The asymptotic approach is then applied to explain the structure of the bow flow in connection with experimental observations and numerical simulations.

The motion of an initially quiescent, incompressible, stratified and/or rotating uid of semi-infinite extent due to surface forcing is considered. The stratification parameter N and the Coriolis parameter f are constant but arbitrary and all possible combinations are considered, including N = 0 (rotating homogeneous fluid), f = 0 (non-rotating stratified fluid) and the special case N = f. The forcing is suction or pumping at an upper rigid surface and the response consists of geostrophic flows and inertial-internal waves. The response to impulsive point forcings (Green's functions) is contrasted with the response to finite-sized circularly symmetric impulsive forcings. Early-time and large-time behaviour are studied in detail. At early times transient internal waves change the vortices that are created by pumping/suction at the surface. The asymptotically remaining vortices are determined, a simple expression for what fraction of the initial energy is converted into internal waves is derived, as well as wave energy fluxes and the dependence of the flux direction on the value of N/f. The internal wave field is to leading order in time a distinct pulse, and rules for the arrival time of the pulse, its amplitude, its motion along a ray of constant frequency and decay with time, are given for the far field. A simple formula for the total wave energy distribution as a function of frequency is derived for when all waves have propagated away from the forcing.

We examine the motion of a two-layer gravity current, composed of two fluids of different viscosity and density, as it propagates through a model porous layer. We focus on two specific situations: first, the case in which each layer of fluid has finite volume, and secondly, the case in which each layer is supplied by a steady maintained flux. In both cases, we find similarity solutions which describe the evolution of the flow. These solutions illustrate how the morphology of the interface between the two layers of fluid depends on the viscosity, density and volume ratios of the two layers. We show that in the special case that the viscosity ratio of the upper to lower layers, V, satisfies V = (1 + F)/(1 + RF) where F and R are respectively the ratios of the volume and buoyancy of the lower layer to those of the upper layer, then the ratio of layer depths is the same at all points. Furthermore, we show that for V > (<)(1 + F)/(1 + RF), the lower (upper) layer advances ahead of the upper (lower) layer. We also present some new laboratory experiments on two-layer gravity currents, using a Hele-Shaw cell, and show that these are in accord with the model predictions. One interesting prediction of the model, which is confirmed by the experiments, is that for a finite volume release, if the viscosity ratio is sufficiently large, then the less-viscous layer separates from the source. We extend the model to describe the propagation of a layer of fluid which is continuously stratified in either density or viscosity, and we briefly discuss application of the results for modelling various two-layer gravity-driven flows in permeable rock.

A theory for fully developed turbulent pipe and channel flows is proposed which extends the classical analysis to include the effects of finite Reynolds number. The proper scaling for these flows at finite Reynolds number is developed from dimensional and physical considerations using the Reynolds-averaged Navier–Stokes equations. In the limit of infinite Reynolds number, these reduce to the familiar law of the wall and velocity deficit law respectively.

The fact that both scaled profiles describe the entire flow for finite values of Reynolds number but reduce to inner and outer profiles is used to determine their functional forms in the ‘overlap’ region which both retain in the limit. This overlap region corresponds to the constant, Reynolds shear stress region (30 < y+ < 0.1R+ approximately, where R+ = u*R/v). The profiles in this overlap region are logarithmic, but in the variable y + a where a is an offset. Unlike the classical theory, the additive parameters, Bi, Bo, and log coefficient, 1/κ, depend on R+. They are asymptotically constant, however, and are linked by a constraint equation. The corresponding friction law is also logarithmic and entirely determined by the velocity profile parameters, or vice versa.

It is also argued that there exists a mesolayer near the bottom of the overlap region approximately bounded by 30 < y+ < 300 where there is not the necessary scale separation between the energy and dissipation ranges for inertially dominated turbulence. As a consequence, the Reynolds stress and mean flow retain a Reynolds number dependence, even though the terms explicitly containing the viscosity are negligible in the single-point Reynolds-averaged equations. A simple turbulence model shows that the offset parameter a accounts for the mesolayer, and because of it a logarithmic behaviour in y applies only beyond y+ > 300, well outside where it has commonly been sought.

The experimental data from the superpipe experiment and DNS of channel flow are carefully examined and shown to be in excellent agreement with the new theory over the entire range 1.8 × 102 < R+ < 5.3 × 105. The Reynolds number dependence of all the parameters and the friction law can be determined from the single empirical function, H = A/(ln R+)α for α > 0, just as for boundary layers. The Reynolds number dependence of the parameters diminishes very slowly with increasing Reynolds number, and the asymptotic behaviour is reached only when R+ [Gt ] 105.

An analytical model is presented for the direct initiation of gaseous detonations by a blast wave. For stable or weakly unstable mixtures, numerical simulations of the spherical direct initiation event and local analysis of the one-dimensional unsteady reaction zone structure identify a competition between heat release, wave front curvature and unsteadiness. The primary failure mechanism is found to be unsteadiness in the induction zone arising from the deceleration of the wave front. The quasi-steady assumption is thus shown to be incorrect for direct initiation. The numerical simulations also suggest a non-uniqueness of critical energy in some cases, and the model developed here is an attempt to explain the lower critical energy only. A critical shock decay rate is determined in terms of the other fundamental dynamic parameters of the detonation wave, and hence this model is referred to as the critical decay rate (CDR) model. The local analysis is validated by integration of reaction-zone structure equations with real gas kinetics and prescribed unsteadiness. The CDR model is then applied to the global initiation problem to produce an analytical equation for the critical energy. Unlike previous phenomenological models of the critical energy, this equation is not dependent on other experimentally determined parameters and for evaluation requires only an appropriate reaction mechanism for the given gas mixture. For different fuel–oxidizer mixtures, it is found to give agreement with experimental data to within an order of magnitude.

The two-way coupling mechanisms in particle-laden mixing layers are investigated, with and without particle settling, and with an emphasis on the resulting modifications to the fluid vorticity field. The governing equations are interpreted with respect to the production and cancellation of vorticity. These mechanisms are shown to be related to the misalignment of the concentration gradient and the slip velocity, as well as to the difference in fluid and particle vorticities. Preliminary insight into the physics is obtained from an analysis of the unidirectional base flow. For this model problem, the conditions are established under which the particle velocity remains a single-valued function of space for all times. The resulting simplified set of two-way-coupled equations governing the vorticity of the fluid and particulate phases, respectively, is solved numerically. The formation of a decaying travelling wave solution is demonstrated over a wide range of parameters. Interestingly, the downward propagation of the fluid vorticity field is not accomplished through convection, but rather by the production and loss of vorticity on opposite sides of the mixing layer. For moderate settling velocities, the simulation results reveal an optimal coupling mechanism between the fluid and particle vorticities at intermediate values of the mass loading parameter. For large settling velocities and intermediate mass loadings, more than one local maximum is seen to evolve in the vorticity field. A scaling law for the downward propagation rates of the vorticity fronts is derived.

Two-dimensional particle-laden mixing layers are investigated by means of a mixed Lagrangian–Eulerian approach which is based on the vorticity variable. For uniformly seeded mixing layers, the simulations confirm some of the features observed by Druzhinin (1995b) for the model problem of a two-way-coupled particle-laden Stuart vortex, as well as by Dimas & Kiger (1998) in a linear stability analysis. For small values of the Stokes number, a mild destabilization of the mixing layer is observed. At moderate and large Stokes numbers, on the other hand, the transport of vorticity from the braids into the core of the evolving Kelvin–Helmholtz vortices is seen to be slowed by the two-way coupling effects. As a result, the particle ejection from the vortex cores is weakened. For constant mass loadings, the two-way coupling effects are strongest at intermediate Stokes number values. For moderately large Stokes numbers, the formation of two bands of high particle concentration is observed in the braids, which reflects the multi-valued nature of the particle velocity field. For mixing layers in which only one stream is seeded, the particle concentration gradient across the mixing layer leads to strong vorticity production and loss, which results in an effective net motion of the vortex in the flow direction of the seeded stream. Under particle settling, the vortex propagates downward as well. For the parameter range explored here, its settling velocity agrees well with the scaling law derived from the unidirectional flow analysis.

This work uses direct numerical simulations of time evolving annular mixing layers, which correspond to the early development of round jets, to study compressibility effects on turbulence in free shear flows. Nine cases were considered with convective Mach numbers ranging from Mc = 0.1 to 1.8 and turbulence Mach numbers reaching as high as Mt = 0.8.

Growth rates of the simulated mixing layers are suppressed with increasing Mach number as observed experimentally. Also in accord with experiments, the mean velocity difference across the layer is found to be inadequate for scaling most turbulence statistics. An alternative scaling based on the mean velocity difference across a typical large eddy, whose dimension is determined by two-point spatial correlations, is proposed and validated. Analysis of the budget of the streamwise component of Reynolds stress shows how the new scaling is linked to the observed growth rate suppression. Dilatational contributions to the budget of turbulent kinetic energy are found to increase rapidly with Mach number, but remain small even at Mc = 1.8 despite the fact that shocklets are found at high Mach numbers. Flow visualizations show that at low Mach numbers the mixing region is dominated by large azimuthally correlated rollers whereas at high Mach numbers the flow is dominated by small streamwise oriented structures. An acoustic timescale limitation for supersonically deforming eddies is found to be consistent with the observations and scalings and is offered as a possible explanation for the decrease in transverse lengthscale.

The mixing of fuel and oxidizer in a mixing layer between high-speed streams is important in many applications, especially air-breathing propulsion systems. The details of this process in a turbulent annular mixing layer are studied with direct numerical simulation. Convective Mach numbers of the simulations range from Mc = 0.1 to Mc = 1.8. Visualizations of the scalar field show that at low Mach numbers large intrusions of nearly pure ambient or core fluid span the mixing region, whereas at higher Mach numbers these intrusions are suppressed. Increasing the Mach number is found to change the mixture fraction probability density function from non-marching to marching and the mixing efficiency from 0.5 at Mc = 0.1 to 0.67 at Mc = 1.5. Scalar concentration fluctuations and the axial velocity fluctuations become highly correlated as the Mach number increases and a suppressed role of pressure in the axial momentum equation is found to be responsible for this. Anisotropy of scalar flux increases with Mc, and is explained via the suppression of transverse turbulence lengthscale.

Thermal convection in a horizontal water layer being cooled from above and absorbing solar radiation is simulated numerically at the Prandtl number Pr = 7. Three different regimes arising are investigated here. The first is characterized by intermittent convection, the second by steady-state convection, and the third is convection free. The transitions occur at different values of J0/Q, the ratio of downward solar-radiation flux just below the surface to heat flux through the interface (assumed to be constant), but at almost the same Rayleigh number. The generalized heat-conduction law is found to be valid.

For the purpose of studying the spectral properties of energy transfer between large and small scales in high-Reynolds-number turbulence, we measure the longitudinal subgrid-scale (SGS) dissipation spectrum, defined as the co-spectrum of the SGS stress and filtered strain-rate tensors. An array of four closely spaced X-wire probes enables us to approximate a two-dimensional box filter by averaging over different probe locations (cross-stream filtering) and in time (streamwise filtering using Taylor's hypothesis). We analyse data taken at the centreline of a cylinder wake at Reynolds numbers up to Rλ ∼ 450. Using the assumption of local isotropy, the longitudinal SGS stress and filtered strain-rate co-spectrum is transformed into a radial co-spectrum, which allows us to evaluate the spectral eddy viscosity, v(k, kΔ). In agreement with classical two-point closure predictions, for graded filters, the spectral eddy viscosity deduced from the box-filtered data decreases near the filter wavenumber kΔ. When using a spectral cutoff filter in the streamwise direction (with a box-filter in the cross-stream direction) a cusp behaviour near the filter scale is observed. In physical space, certain features of a wavenumber-dependent eddy viscosity can be approximated by a combination of a regular and a hyper-viscosity term. A hyper-viscous term is also suggested from considering equilibrium between production and SGS dissipation of resolved enstrophy. Assuming local isotropy, the dimensionless coefficient of the hyper-viscous term can be related to the skewness coefficient of filtered velocity gradients. The skewness is measured from the X-wire array and from direct numerical simulation of isotropic turbulence. The results show that the hyper-viscosity coefficient is negative for graded filters and positive for spectral filters. These trends are in agreement with the spectral eddy viscosity measured directly from the SGS stress–strain rate co-spectrum. The results provide significant support, now at high Reynolds numbers, for the ability of classical two-point closures to predict general trends of mean energy transfer in locally isotropic turbulence.

We consider the flow of a dilute suspension of equisized solid spheres in a viscous fluid. The viscosity of such a suspension is dependent on the volume fraction, c, of solid particles. If the particles are perfectly smooth, then solid spheres will not come into contact, because lubrication forces resist their approach. In this paper, however, we consider particles with microscopic surface asperities such that they are able to make contact. For straining motions we calculate the O(c2) coefficient of the resultant viscosity, due to pairwise interactions. For shearing motions (for which the viscosity is undetermined because of closed orbits on which the probability distribution is unknown) we calculate the c2 contribution to the normal stresses N1 and N2. The viscosity in strain is shown to be slightly lower than that for perfectly smooth spheres, though the increase in the O(c) term caused by the increased effective radius due to surface asperities will counteract this decrease. The viscosity decreases with increasing contact friction coefficient. The normal stresses N1 and N2 are zero if the surface roughness height is less than a critical value of 2.11 × 10−4 times the particle radius, and then become negative as the roughness height is increased above this value. N1 is larger in magnitude than N2.

We consider the effect of steady, three-dimensional cellular convective fields impressed upon the moving front of a dilute binary alloy in directional solidification. The flows have length scales longer than the characteristic lengths of the morphological instability. A Floquet problem with multiple degrees of freedom in space governs the interfacial dynamics and determines the morphological patterns and marginal stability boundaries. In the cases of weak flows the induced patterns are superpositions of rolls modulated by the forced flows. When the flows are strong, the instability becomes spatially localized and confined at inward flow-stagnation regions on the front. Numerical computations and the WKB method are used to solve the eigenvalue problems, showing various localized states depending on the structures of the imposed flows.