In the Lagrangian representation, the problem of advection of a passive marker particle by a prescribed flow defines a dynamical system. For two-dimensional incompressible flow this system is Hamiltonian and has just one degree of freedom. For unsteady flow the system is non-autonomous and one must in general expect to observe chaotic particle motion. These ideas are developed and subsequently corroborated through the study of a very simple model which provides an idealization of a stirred tank. In the model the fluid is assumed incompressible and inviscid and its motion wholly two-dimensional. The agitator is modelled as a point vortex, which, together with its image(s) in the bounding contour, provides a source of unsteady potential flow. The motion of a particle in this model device is computed numerically. It is shown that the deciding factor for integrable or chaotic particle motion is the nature of the motion of the agitator. With the agitator held at a fixed position, integrable marker motion ensues, and the model device does not stir very efficiently. If, on the other hand, the agitator is moved in such a way that the potential flow is unsteady, chaotic marker motion can be produced. This leads to efficient stirring. A certain case of the general model, for which the differential equations can be integrated for a finite time to produce an explicitly given, invertible, area-preserving mapping, is used for the calculations. The paper contains discussion of several issues that put this regime of chaotic advection in perspective relative to both the subject of turbulent advection and to recent work on critical points in the advection patterns of steady laminar flows. Extensions of the model, and the notion of chaotic advection, to more realistic flow situations are commented upon.

The local interaction of an oblique shock wave with an unseparated turbulent boundary layer at a shallow two-dimensional compression corner is described by asymptotic expansions for small values of the non-dimensional friction velocity and the flow turning angle. It is assumed that the velocity-defect law and the law of the wall, adapted for compressible flow, provide an asymptotic representation of the mean velocity profile in the undisturbed boundary layer. Analytical solutions for the local mean-velocity and pressure distributions are derived in supersonic, hypersonic and transonic small-disturbance limits, with additional intermediate limits required at distances from the corner that are small in comparison with the boundary-layer thickness. The solutions describe small perturbations in an inviscid rotational flow, and show good agreement with available experimental data in most cases where effects of separation can be neglected. Calculation of the wall shear stress requires solution of the boundary-layer momentum equation in a sublayer which plays the role of a new thinner boundary layer but which is still much thicker than the wall layer. An analytical solution is derived with a mixing-length approximation, and is in qualitative agreement with one set of measured values.

The Zakharov integral equation for surface gravity waves is modified to include higher-order (quintet) interactions, for water of constant (finite or infinite) depth. This new equation is used to study some aspects of class I (4-wave) and class II (5-wave) instabilities of a Stokes wave.

The spin-up of a two-layer fluid in a cylinder with a free surface and with a thin lower layer is examined in the laboratory. It is found that when the increase in rotation rate of the cylinder is large enough the radial outflow in the viscous boundary layer on the tank bottom is sufficient to cause the interface to descend near the centre of the tank and to intersect the bottom. The intersection between the interface and the bottom produces a front between stratified (two-layer) fluid and a central region (called the bare spot) in which the upper layer is in direct contact with the bottom. It is observed that the radius of the circular bare spot increases until the lower layer is spun up. Observations of the maximum size of the bare spot are compared with a theoretical calculation in which it is assumed that the lower layer acquires the new angular velocity of the container and where viscous coupling between the layers is neglected. An expansion in F, the upper-layer Froude number, gives good agreement with the observations.

At larger times the circular front is observed to be unstable to frontal waves which appear to gain energy via baroclinic instability from the sloping density interface. At large amplitude these waves ‘break’, producing regions of closed streamlines in the upper layer. The shape of the bare spot is severely distorted by these waves and the associated motions. The effect of the bottom stress on the spin-up of the upper layer is found to be limited to the bare spot where it is in direct contact with the bottom. Some comments are made on the formation and decay of fronts in the benthic boundary layer of the ocean.

The local-energy-transfer (LET) theory (McComb 1978) was used to calculate freely decaying turbulence for four different initial spectra at low-to-moderate values of microscale Reynolds numbers (Rλ up to about 40). The results for energy, dissipation and energy-transfer spectra and for skewness factor all agreed quite closely with the predictions of the well-known direct-interaction approximation (DIA: Kraichnan 1964). However, LET gave higher values of energy transfer and of evolved skewness factor than DIA. This may be related to the fact that LET yields the k−5/3 law for the energy spectrum at infinite Reynolds number.

The LET equations were then integrated numerically for decaying isotropic turbulence at high Reynolds number. Values were obtained for the wavenumber spectra of energy, dissipation rate and inertial-transfer rate, along with the associated integral parameters, at an evolved microscale Reynolds number Rλ of 533. The predictions of LET agreed well with experimental results and with the Lagrangian-history theories (Herring & Kraichnan 1979). In particular, the purely Eulerian LET theory was found to agree rather closely with the strain-based Lagrangian-history approximation; and further comparisons suggested that this agreement extended to low Reynolds numbers as well.

This paper considers the temporal evolution of two-dimensional Rayleigh–Bénard convection in a shallow fluid layer of aspect ratio 2L ([Gt ] 1) confined laterally by rigid sidewalls. Recent studies by Cross et al. (1980, 1983) have shown that for Rayleigh numbers in the range R = R0 + O(L−1) (where R0 is the critical Rayleigh number for the corresponding infinite layer) there exists a class of finite-amplitude steady-state ‘phase-winding’ solutions which correspond physically to the possibility of an adjustment in the number of rolls in the container as the local value of the Rayleigh number is varied. It has been shown (Daniels 1981) that in the temporal evolution of the system the final lateral positioning of the rolls occurs on the long timescale t = O(L2) when the phase function which determines the number of rolls in the system satisfies a one-dimensional diffusion equation but with novel boundary conditions that represent the effect of the sidewalls. In the present paper this system is solved numerically in order to determine the precise way in which the roll pattern adjusts after a change in the Rayleigh number of the system. There is an interesting balance between, on the one hand, a tendency for the number of rolls to change by the least number possible and, on the other, a tendency for the even or odd nature of the initial configuration to be preserved during the transition. In some cases this second property renders the natural evolution susceptible to arbitrarily small external disturbances, which then dictate the form of the final roll pattern.

The complete transition involves an analysis of the motion on three timescales, a conductive scale t = O(1), a convective growth scale t = O(L) and a convective diffusion scale t = O(L2).

A wind-tunnel investigation was made of the mechanism of separation of a two-dimensional turbulent boundary layer on a convex wall. The flow field was observed visually by using a large number of smoke wires arranged in various ways. Statistical quantities were obtained by newly developed direction-sensitive hot-wire probes and flow-direction meters. Smoke pictures show localized backflow spots in the separation region. They occur intermittently, grow downstream, merge with each other and eventually cover the whole flow field. Measurements of instantaneous flow direction show that velocity fluctuations in the separation region are strongly three-dimensional. The backflow factor, which is defined as the fraction of time of occurrence of backflow, is used for the quantitative description of the separation region. The role of large-scale ordered motions in the turbulent separation was investigated by use of the conditional sample and average technique. It was confirmed that a localized backflow is initiated by a large-scale low-speed lump of fluid which travels downstream.

The problem of fluid transport by cilia in a circular cylinder is investigated. The discrete-cilia approach is used in building the model, using the Green function due to an infinite periodic Stokeslet array in a pipe. Two different expressions are obtained for the Green function, one via a residue method and the other using the Poisson summation formula each amenable for computation in a different region. Interaction of the Stokeslets is investigated to see how, as distance decreases, interaction changes from initially separated closed vortices to a continuous flow. The singular integral equations for the forces in this model are now replaced by non-singular equations, thus overcoming the numerical difficulties in earlier works. It is found that in the pipe core the flow is time-independent and varies between a plug flow and a negative parabolic profile, in the pumping range. These results are seen to be local results due to the near field. Streamlines in the sublayer show eddies near the cilia bases blending into a uniform flow near the cilia tips.

The propagation of surface waves – that is ‘third’ sound – on superfluid helium is considered. The fluid is treated as a continuum, using the two-fluid model of Landau, and incorporating the effects of healing, relaxation, thermal conductivity and Newtonian viscosity. Under the assumptions only of a small amplitude and long wavelength, a linear theory is developed which includes some discussion of the matching to the outer regions of the vapour. This results in a comprehensive propagation speed for linear waves, although a few properties of the flow are left undetermined at this order. A nonlinear theory is then outlined (without dwelling on the details) which leads to the Burgers equation in an appropriate far field, and enables the leading-order theory to be concluded.

Some numerical results, for two temperatures, are presented by first recording the Helmholtz free energy as a polynomial in densities. Owing to the inaccuracies inherent in this procedure, only the equilibrium state can be satisfactorily reproduced, but this is sufficient to predict the onset phenomenon. The propagation speed, as a function of film thickness, is roughly estimated by using the earlier results of Johnson (without healing) suitably doctored to incorporate the new computed superfluid density distributions. The looked-for reduction in the predicted speeds is evident, but the magnitude of this reduction is too large for very thin films. However, it is hoped that the analytical results presented here will prove more effective when a complete and accurate description of the Helmholtz free energy is available.

Application of Landau's ideas to the theory of weakly nonlinear instabilities shows that the amplitude of the unstable modes behaves as the square root of the reduced control parameter ε, its critical value being ε = 0. When applied to cellular structures the theory has been improved by taking into account the slow spatial variations of the amplitude and phase of the unstable modes. Until now the case of thermo-convective instabilities in high vertical channels has not been studied using this approach. In high vertical structures the nonlinear terms disappear in the limit of an infinite height, and the supercritical behaviour requires a specific treatment. It differs from the standard analysis valid for the case of fluid layers of infinite horizontal extent, where the nonlinearities and the finite-size effects are disconnected. In the limit of high aspect ratios (height [Gt ] horizontal extent) we have derived an amplitude equation for convective systems where the nonlinear terms contain derivatives at the lowest order. As a consequence the amplitude equation cannot be put into a variational form and the stability of the stationary solutions cannot be deduced from an ordering in decreasing values of a Lyapunov functional.

The particle trajectories of nonlinear capillary waves are derived. The properties of the surface and subsurface particles are presented in exact analytic form, up to and including the highest wave. It is found that the orbits of the steeper waves are neither circular nor closed. For the highest wave, a particle moves through a distance [X] equal to 7.99556 λ in one orbit, where λ is the wavelength. It moves with an average horizontal drift velocity U equal to 0.88883c, where c is the phase speed of the wave. In addition, the subsurface particles (at depths nearly three-quarters that of the wavelength) move at speeds up to one-tenth that of surface particles.

The dynamics of vortices subject to stretching by a uniform plane straining flow is studied asymptotically and by means of a new class of exact solutions. The asymptotic analysis treats the stretched Burger's vortex sheet for strain rates much greater than the gradient of the sheet strength. It is found that portions of the sheet where the strength density is sufficiently large compared to (viscosity x strain rate)½ will collapse to form concentrated vortices. The exact solutions describe uniform vortices of elliptical cross-section in inviscid fluid subject to stretching parallel to their axes. These solutions complement the description of vortex collapse found by asymptotic methods. The relevance of these results stems from the prevalence of vortex structures subject to strain in turbulent flows.

An experimental study of a high-Reynolds-number turbulent wake in supersonic flow is performed using space and space–time correlation measurements by means of hot-wire anemometry. The correlations for the streamwise component of the mass-flux fluctuations are given for six stations starting from the trailing edge down to the asymptotic part. The validity of the Taylor's hypothesis is tested, the convection velocities are determined and the downstream evolution of the optimum space–time correlation is given; the frequency spectra are discussed and the integral lengths are analysed. Finally, the three-dimensional isocorrelation surfaces are given at the six test stations and discussed in relation to classical incompressible-flow results. The downstream evolution of the correlations shows that the two sides of the wake are statistically independent near the trailing edge, and a statistical link is gradually established during the wake development. A three-zonal description of wakes generated by fully developed turbulent boundary layers applies as well for mean quantities (velocity, width) as for turbulence correlations. In the near-wake region the overall structure of the isocorrelation curves is close to that observed in turbulent boundary layers in incompressible flows; some low-frequency phenomena are observed in this region. In the latest part of the wake, an asymptotic state is reached for all the correlation characteristics; the final state reached is not explained by the double-roller-eddy model established for lower-Reynolds-number wakes; it appears that wakes generated by fully turbulent boundary layers behave quite differently from initially laminar wakes, and new turbulent structure models for high-Reynolds-number wakes are to be devised.

The shock-wave structure close to a wall in pure argon and binary mixtures of noble gases (argon–helium, xenon–helium) is investigated experimentally and numerically in the shock-wave Mach-number range 2·24 ≤ Ms ≤ 9.21. Measured and calculated density profiles are compared, and some conclusions are drawn about the accommodation at the wall and the intermolecular force potential.

For binary gas mixtures only a few results are presented. Weak argon signals of the electron-beam-luminescence method on the experimental side and the computer time needed for the numerical simulation allowed the treatment of a few parameter combinations only.

The asymptotic theory of a high-aspect-ratio wing in an incompressible flow is generalized to an oscillating lifting surface with a curved centreline in the domain where the reduced frequency based on the half-span is of order unity. The formulation allows applications to lightly loaded models of lunate-tail swimming and ornithopter flight, provided that the heaving displacement does not far exceed the mean wing chord. The analysis include the quasi-steady limit, in which the crescent-moon wing problem considered earlier by Thurber (1965) is solved and several aerodynamic properties of swept wings are explained.

Among the important three-dimensional and unsteady effects are corrections for the centreline curvature and for the spanwise components of the locally shed vortices. Comparison of the lift distributions obtained for model lunate tails with data computed from the doublet-lattice method (Albano & Rodden 1969) lends support to the asymptotic theory.

Two-dimensional steady inviscid flow past an inclined flat plate with a forward-facing flap attached to the rear edge is considered for the case when a vortex sheet separates from the leading edge of the flat plate and reattaches at the leading edge of the flap, with uniform vorticity distributed between the vortex sheet and the body. Solutions are found for a particular geometry and a range of values of the vorticity. The method used to calculate the flow is an extension of a free-streamline method widely used in cases where the velocity is a constant on the separating streamline.

A method for the Monte Carlo simulation, by digital computer, of the evolution of a colliding and coagulating population of suspended particles is described. Collision mechanisms studied both separately and in combination are: Brownian motion of the particles, and laminar and isotropic turbulent shearing motions of the suspending fluid. Steady-state distributions are obtained by adding unit-size particles at a constant rate and removing all particles once they reach a preset maximum volume. The resulting size distributions are found to agree with those obtained by dimensional analysis (Hunt 1982).

Hunt (1982) and Friedlander (1960a, b) used dimensional analysis to derive expressions for the steady-state particle-size distribution in aerosols and hydrosols. Their results were supported by the Monte Carlo simulation of a non-interacting coagulating population of suspended spherical particles developed by Pearson, Valioulis & List (1984). Here the realism of the Monte Carlo simulation is improved by accounting for the modification to the coagulation rate caused by van der Waals', electrostatic and hydrodynamic forces acting between particles. The results indicate that the major hypothesis underlying the dimensional reasoning, that is, collisions between particles of similar size are most important in determining the shape of the particle size distribution, is valid only for shear-induced coagulation. It is shown that dimensional analysis cannot, in general, be used to predict equilibrium particle-size distributions, mainly because of the strong dependence of the interparticle force on the absolute and relative size of the interacting particles.

Spanwise structures in a two-dimensional reattaching separated flow were studied using multisensor hot-wire anemometry techniques. The results of these measurements strongly support the existence and importance of large-scale vortices in both the separated and reattached regions of this flow. Upstream of reattachment, vortex pairings are indicated and the spanwise structures attain correlation scales closely comparable to previously measured mixing-layer vortices. These large-scale vortices retain their organization far downstream of the reattachment region. However, pairing interactions appear to be strongly inhibited in this region. It is suggested that large-scale vortex dynamics are primarily responsible for some of the important time-averaged features of this flow. Notably, the reduction of turbulence energy in the reattachment region and the slow transition of the mean flow downstream of reattachment are attributed to effects associated with these vortices.

A simple mathematical model is constructed to describe the regime of flow, extending over a wide range of values of Taylor number, in which turbulent Taylor–Couette flow in the annular region between two coaxial circular cylinders is characterized by the coexistence of steady coherent motion on two widely separated scales. These scales of motion, corresponding to the gap width of the annular region and to a boundary-layer thickness, are each identified as the consequence of a centrifugal instability, and are described as Taylor vortices and Görtler vortices respectively.

The assumption that both scales of motion are near marginal stability gives a closure to a pair of coupled eigenvalue problems, and the results of a linear analysis are shown to be in good agreement with many features of experimental observations.