The mean lift coefficient of a circular cylinder executing rotary motions in a uniform flow is investigated. These motions include steady rotation, and rotary oscillations with a net rotation rate. Results for the steadily rotating cylinder show that for a given rotation rate, larger cylinder aspect ratios yield higher lift coefficients. It was also found that the addition of forced rotary oscillations to the steady rotation of the cylinder increases the lift coefficient in the cases where the wake would normally be separated in the steadily rotating case, but decreases it otherwise. In addition, a method for estimating the mean lift of a rotating cylinder is presented. Estimates based on this method compare favourably with similar data published for steadily rotating cylinders.

Unsteady viscous flow over irregular and fractal walls is discussed, and the flow generated by the longitudinal and transverse vibrations of an infinite periodic two-dimensional wall with cylindrical grooves is considered in detail. The behaviour of the Stokes layer and the functional dependence between the drag force and the frequency are illustrated in a broad band of frequencies for walls with sinusoidal corrugations and a family of walls with triangular asperities leading to fractal shapes. It is shown that, in the case of longitudinal oscillations, the drag force on a fractal wall with self-similar structure exhibits a power-law dependence on the frequency with an exponent that is related to the fractal dimension of the microstructure expressing the gain in surface area with increasing spatial resolution. Numerical evidence suggests that, in the case of transverse oscillations, the dissipative component of the drag force may show a power-law dependence on the frequency, but the exponent is not directly related to the geometry of the microstructure. The significance of these results on the behaviour of the drag force on walls with three-dimensional irregularities is discussed.

Motivated by the observed potential vorticity structure of the stratospheric polar vortex, we study the dynamics of linear and nonlinear waves on a zonal vorticity interface in a two-dimensional barotropic flow on the surface of a sphere (interfacial Rossby waves). After reviewing the linear problem, we determine, with the help of an iterative scheme, the shapes of steadily propagating nonlinear waves; a stability analysis reveals that they are (nonlinearly) stable up to very large amplitude.

We also consider multi-vortex equilibria on a sphere: we extend the results of Thompson (1883) and show that a (latitudinal) ring of point vortices is more unstable on the sphere than in the plane; notably, no more than three point vortices on the equator can be stable. We also determine the shapes of finite-area multi-vortex equilibria, and reveal additional modes of instability feeding off shape deformations which ultimately result in the complex merger of some or all of the vortices.

We discuss two specific applications to geophysical flows: for conditions similar to those of the wintertime terrestrial stratosphere, we show that perturbations to a polar vortex with azimuthal wavenumber 3 are close to being stationary, and hence are likely to be resonant with the tropospheric wave forcing; this is often observed in high-resolution numerical simulations as well as in the ozone data. Secondly, we show that the linear dispersion relation for interfacial Rossby waves yields a good fit to the phase velocity of the waves observed on Saturn's ‘ribbon’.

The structure of the intense-vorticity regions is studied in numerically simulated homogeneous, isotropic, equilibrium turbulent flow fields at four different Reynolds numbers, in the range Reλ = 35–170. In accordance with previous investigators this vorticity is found to be organized in coherent, cylindrical or ribbon-like, vortices (‘worms’). A statistical study suggests that they are simply especially intense features of the background, O(ω′), vorticity. Their radii scale with the Kolmogorov microscale and their lengths with the integral scale of the flow. An interesting observation is that the Reynolds number γ/ν, based on the circulation of the intense vortices, increases monotonically with Reλ, raising the question of the stability of the structures in the limit of Reλ → ∞. Conversely, the average rate of stretching of these vortices increases only slowly with their peak vorticity, suggesting that self-stretching is not important in their evolution. One- and two-dimensional statistics of vorticity and strain are presented; they are non-Gaussian and the behaviour of their tails depends strongly on the Reynolds number. There is no evidence of convergence to a limiting distribution in this range of Reλ, even though the energy spectra and the energy dissipation rate show good asymptotic properties in the higher-Reynolds-number cases. Evidence is presented to show that worms are natural features of the flow and that they do not depend on the particular forcing scheme.

The steady flow past the bow of a two-dimensional ship in water of infinite depth is considered. The ship is assumed to be a semi-infinite flat-bottomed body terminated by a face inclined at an angle β with the horizontal. The spray is modelled by a layer of water rising along the bow and falling back as a jet. A series truncation method is used to solve the fully nonlinear problem numerically. It is shown that for a prescribed value of β, there is a one-parameter family of solutions. Values of the drag and of the jet thickness are presented for different values of β.

The flow of a periodic suspension of two-dimensional viscous drops in a closed channel that is bounded by two parallel plane walls executing relative motion is studied numerically using the method of interfacial dynamics. Ordered suspensions where at the initial instant the drops are arranged in several layers on a hexagonal lattice are considered for a variety of physical conditions and geometrical configurations. It is found that there exists a critical capillary number below which the suspensions exhibit stable periodic motion, and above which the drops elongate and tend to coalesce, altering the topology of the initial configuration. At sufficiently large volume fractions, a minimum drop capillary number exists below which periodic motion is suppressed owing to the inability of the drops to deform and bypass other neighbouring drops in adjacent layers. This feature distinguishes the motion of dense emulsions from that of foam. The effects of capillary number, viscosity ratio, volume fraction of the dispersed phase, lattice geometry, and instantaneous drop shape, on the effective stress tensor of the suspension are illustrated and the results are discussed with reference to theories of foam. Two simulations of a random suspension with 12 drops per periodic cell are performed, and the salient features of the motion are identified and discussed. These include pairing, tripling, and higher-order interactions among intercepting drops, cluster formation and destruction, and drop migrations away from the walls. The macroscopic features of the flow of random suspensions are shown to be significantly different from those of ordered suspensions and quite independent of the initial condition. The general behaviour of suspensions of liquid drops is compared to that of suspensions of rigid spherical particles, and some differences are discussed.

The physical- and orientation-space transport of non-spherical, generally non-neutrally buoyant, Brownian particles in unbounded homogeneous shear flows is analysed with the goal of studying the respective effects of the orientational degrees of freedom of such particles upon their sedimentation and dispersion rates. In particular, the present contribution concentrates on the interaction between the Taylor dispersion mechanism (arising from coupling between the orientational dependence of the particle's translational velocity and the stochastic sampling of the orientation space via rotary Brownian diffusion) and the shear velocity field.

Making use of a recent extension of generalized Taylor dispersion theory to homogeneous (unbounded) shear flows, the mean transport process in physical space is modelled by a convection–diffusion problem characterized by a pair of constant phenomenological coefficients, provided that the eigenvalues of the (constant) undisturbed velocity gradient are purely imaginary. The latter phenomenological coefficients – namely, U*, the average ‘slip velocity’ vector (of the particles relative to the ambient fluid), and D*, the dispersivity dyadic or, equivalently, the pair of dyadics $\bar{\bm M}$ and DC (or $\bar{\bm D}^c$), the average mobility and the Taylor (or modified Taylor) dispersivity, respectively – are evaluated both asymptotically (in the respective limits of small and large rotary Peclét numbers) as well as numerically (for arbitrary Peclét numbers).

It is established that (up to a scalar multiplication factor, independent of Peclét number) the anisotropic portion of the average mobility is formally equivalent to the direct diffusive contribution to the particle stress in the context of suspension rheology.

The analysis focuses mainly on the case of simple shear flow. The approximate calculation in the limit of large Peclét numbers, Pe [Gt ] 1, which makes extensive use of the ‘natural coordinates’ along Jeffery orbits previously introduced by Leal & Hinch, verifies that, if the external force is non-orthogonal to the direction along which (undisturbed) fluid velocity variations occur, two of the eigenvalues of Dc are proportional to Pe; moreover, one of these O(Pe) eigenvalues is negative. When the external force is parallel to the latter direction, the negative eigenvalue corresponds to the principal direction of contraction in the shear velocity field; this thus relates the non-positive nature of DC to the interaction between the Taylor dispersion mechanism and the (deterministic) convection within the shear field.

Explicit results for the variation of the dyadics $\bar{\bm M}$, Dc and $\bar{\bm D}^{c}$ jointly with the respective magnitude of the shear rate and the deviation of the particle geometry from a spherical shape are presented for spheroidal particles. Among other things, it is demonstrated that the proposed definition of the modified Taylor dispersivity coefficient, $\bar{\bm D}^{c}$, does indeed yield a non-negative dyadic.

A study of two-layer quasi-geostrophic vortex flow is performed to determine the effect of a current difference between the layers on a vortex initially extending through both layers. In particular, the conditions under which the vortex can resist being torn by the current difference are examined. The vortex evolution is determined using versions of the contour dynamics and discrete vortex methods which are modified for two-layer quasi-geostrophic flows. The vortex response is found to depend upon the way in which the current difference between the layers is maintained. In the first set of flows studied, the current difference is generated by a (stronger) third vortex in the upper layer located at a large distance from the (weaker) vortex under study. Flows of this type are important for understanding the interactions of vortices of different sizes in geophysical turbulence. A set of flows is also considered in which an ambient geostrophic current difference is produced by a non-uniform background potential vorticity field. In this case, an additional (secondary) flow field about the vortex patch in each layer is generated by redistribution of the ambient potential vorticity field.

It is found that a vortex that initially extends through both layers will undergo a periodic motion, in which the two parts of the initial vortex in the different layers (called the ‘upper’ and ‘lower’ vortices) oscillate about each other, provided that the current difference between the layers is less than a critical value. When the current difference exceeds this critical value, the upper and lower vortices separate permanently and the initial vortex is said to ‘tear’. The effects of various dimensionless parameters that characterize the flow are considered, including the ratio of core radius to internal Rossby radius, the ratio of layer depths and the ratio of the strengths of the upper and lower vortices. These parameters affect both the critical current difference for tearing and the deformation of the vortex cores by their interaction. It is found that for small values of inverse internal Rossby deformation radius, calculations with circular non-deformable vortices (convected at their centrepoints) give results in good agreement with the contour dynamics simulations, since the vortex deformation is small. The results of the study will be useful in determining the conditions under which large geophysical vortex structures, such as cyclones and ocean rings, can extend to large heights (depths) even though the mean winds (currents) in the ambient flow change significantly along the vortex length.

Asymptotic expressions for the onset of convection in a horizontal fluid layer of finite extent heated from below and rotating about a vertical axis are derived in the limit of large rotation rates in the case of stress-free upper and lower boundaries. In the presence of vertical sidewalls, the critical Rayleigh number Rc is much lower than the classical value for an infinitely extended layer. In particular, we find that Rc grows in proportion to τ when the sidewall is insulating, where τ is the dimensionless rotation rate. When the sidewall is infinitely conducting, Rc grows in proportion to $\tau^{\frac{4}{3}}$ as in the case of an infinitely extended layer but with a lower coefficient of proportionality. Numerical results obtained at finite values of τ show good agreement with the asymptotic formulae.

Towing-tank experiments on linearly stratified flow over three-dimensional obstacles of various shapes are described. Particular emphasis is given to the parameter regimes which lead to wave breaking aloft, the most important of which is the Froude number defined by Fh = U/Nh, where U, N and h are the flow speed, the Brunt–Väisälä frequency and the hill height, respectively. The effects of other parameters, principally K (= ND/πU, where D is the fluid depth) and the spanwise and longitudinal aspect ratios of the hill, on wave breaking are also demonstrated. It is shown that the Froude-number range over which wave breaking occurs is generally much more restricted than the predictions of linear (hydrostatic) theories would suggest; nonlinear (Long's model) theories are in somewhat closer agreement with experiment. The results also show that a breaking wave aloft can exist separately from a further recirculating region downstream of the hill under the second lee wave, but that under certain circumstances these can interact to form a massive turbulent zone whose height is much greater than h. Previous theories only give estimates for the upper critical Fh, below which breaking occurs; the experiments also reveal lower critical values, below which there is no wave breaking.

This paper complements the instability theory of frontal waves investigated by Orlanski (1968), and reinterprets the unstable modes obtained. First, the stability of a frontal model is reconsidered by using a matrix method. The major part of Orlanski's (1968) result is verified but some flaws are found in some parameter regions: unstable modes do not exist over the entire Ri–Ro region. Also, the features of the neutral waves in the one-layer subsystems are studied, in order to determine the instability of the full two-layer system. As a result, the unstable mode called the (B)-mode by Orlanski (1968) and suggested by Sakai (1989) to be Rossby-Kelvin instability caused by a resonance between a Rossby wave and a gravity wave, proves to be a geostrophic unstable mode caused by resonance between a Rossby wave and the Rossby-gravity mixed mode. In addition, some of the analytical conclusions about the stability of this frontal model are explained by the features of the neutral waves in the one-layer subsystem.

When small, light solid particles are fluidized by gases it is well known that stable expansion occurs over a finite interval of gas flow beyond the point of minimum fluidization. The existence of such an interval can be predicted from linear stability theory provided the momentum equation for the particles contains a sufficiently large term representing an effective pressure that increases with the concentration of the particles. There is at present some controversy regarding the physical origin of such a term. Some workers attribute it to forces exerted between particles at points of solid–solid contact, while others invoke hydrodynamic mechanisms related to the interaction between the particles and the fluid. In this paper the processes of fluidization and defluidization for fine particles are followed very carefully round complete cycles, starting from zero gas flow and extending to a value at which bubbles appear, then back to zero. The depth of the bed and the pressure drop in the gas traversing it are recorded at each stage, and vertical profiles of the volume fraction of particulate material are determined with a high-resolution gamma-ray densitometer. Similar information is also obtained for sub-cycles extending over more restricted intervals of the gas flow rate. The particles studied are cracking catalyst, with mean diameter 75 μm, and Ottawa sand with mean diameter 154 μm. The results lead to the conclusion that the particle assemblies exhibit yield stresses throughout the range of stable behaviour, and thus are not truly fluidized beds, in the accepted sense. The phenomena observed are such that it is most unlikely that their origin is hydrodynamic. For the particular systems studied we therefore conclude that contact forces are responsible for stabilization.

This paper presents the stochastic properties of orbital velocities of random water waves in intermediate water depth. Both the emergence effect and weak nonlinear effects are studied; the theoretical predictions are compared with measured kinematics and the deviations from linear theory are quantified.

This study includes new ideas in fluid dynamics. An analytic formula for probability distribution for velocities modified by the emergence effect as well as by nonlinearities of the wave motion in intermediate water depth is developed. This probability function gives us the first statistical moment, the second statistical moment for modified velocities in an analytical form, and by numerical integration the third statistical moment for modified velocities.

The theoretical formulae for the statistical moments for surface elevation and for velocities up to third order, with nonlinearities of the motion taken into account, for the case when the emergence effect can be neglected, i.e. below the surface layer, have been developed. This includes a generalized formula for free-surface elevation setdown and calculation of the asymmetry of the horizontal velocity, which is found to be negative in agreement with measurements of Anastasiou et al. (1982b).

From the first statistical moment of the modified horizontal velocity, the mean flux between any two levels in the wave flume may be calculated. When the integration is carried out from the bottom up to + ∞, it leads in approximation to the formula for total mean flux found by Phillips (1960). This agreement with Phillips’ formula encourages one to interpret the positive mean value of horizontal velocities as a ‘real current’. This interpretation also provides a new understanding of the fluid dynamic implications of results presented by Tung (1975).

Theoretical prediction of the measured kinematics has allowed a better estimation of the return flow in the wave flume, and in the vicinity of the mean water level currents in two different directions are noted. Firstly, the emergence effect gives rise to a current at the mean water level in the direction of the wave advance. Secondly, a flow in the opposite direction, interpreted as a return current in the wave flume, is noticed just below that level.

A stability analysis is undertaken to theoretically study the effects of gravity modulation and cross-diffusion on the onset of convection in horizontally unbounded doubly diffusive fluid layers. We investigate the stability of doubly stratified incompressible Boussinesq fluid layers with stress-free and rigid boundaries when the stratification is either imposed or induced by Soret separation. The stability criteria are established by way of Floquet multipliers of the amplitude equations. The topology of neutral curves and stability boundaries exhibits features not found in modulated singly diffusive or unmodulated multiply diffusive fluid layers. A striking feature in gravity-modulated doubly cross-diffusive layers is the existence of bifurcating neutral curves with double minima, one of which corresponds to a quasi-periodic asymptotically stable branch and the other to a subharmonic neutral solution. As a consequence, a temporally and spatially quasi-periodic bifurcation from the basic state is possible, in which case there are two incommensurate critical wavenumbers at two incommensurate onset frequencies at the same Rayleigh number. In some instances, the minimum of the subharmonic branch is more sensitive to small parameter variations than that of the quasi-periodic branch, thus affecting the stability criteria in a way that differs substantially from that of unmodulated layers.

A two-dimensional hot-air jet is investigated experimentally in the transitional regime. The density effect on the near-field behaviour of the initially laminar jet is explored by flow visualization, mean flow measurements and spectral analysis of fluctuating data. It is shown that the broadband amplitude spectra which characterize cold jets become line-dominated for hot jets when the ratio of the jet-exit to the ambient density is below approximately 0.9. Below this critical density ratio the oscillations in the hot jet are shown to be self-excited. That is, the onset of the global oscillations is identified as a Hopf bifurcation and the critical parameter is determined from amplitude spectra and autobicoherence, with the latter proving to be more reliable. Furthermore, the development of three-dimensional structures, which contribute to the jet spreading, is revealed by flow visualization. It is found that, for the parameters investigated, the spreading of the two-dimensional hot jet is not as spectacular as in the axisymmetry case.

In certain geophysical contexts such as lava lakes and mantle convection, a cold, viscous boundary layer forms over a deep pool. The following model problem investigates the buoyant instability of the layer. Beneath a shear-free horizontal boundary, a thin layer (thickness d1) of very viscous fluid overlies a deep layer of less dense, much less viscous fluid; inertia and surface tension are negligible. After the initial unstable equilibrium is perturbed, a long-wave analysis describes the growth of the disturbance, including the nonlinear effects of large amplitude. The results show that nonlinear effects greatly enhance growth, so that initial local maxima in the thickness of the viscous film grow to infinite thickness in finite time, with a timescale 8μ/Δρgd1. In the final catastrophic growth the peak thickness is inversely proportional to the remaining time. (A parallel analysis for fluids with power-law rheology shows similar catastrophic growth.) While the small-slope approximation must fail before this singular time, the failure is only local, and a similarity solution describes how the peaks become downwelling plumes as the viscous film drains away.

Finite-amplitude, forced and free oscillations of capillary bridges are studied. They are characterized by a resonant frequency and a damping rate which, in turn, depend on fluid properties, dimensions of the bridge, gravitational force relative to surface tension and amplitude of the external disturbance. The Navier–Stokes equations are solved numerically using the Galerkin/finite-element methodology for discretization in space and implicit finite differences with adaptive time stepping for discretization in time. It is found that the resonant frequency decreases and the damping rate increases almost linearly with the oscillation amplitude. Their relative changes from their corresponding values at infinitesimal amplitude depend on fluid properties and dimensions of the bridge. Moreover, careful measurement of the resonant frequency and damping rate in a well-controlled experiment may provide quite accurate values for properties of the liquid over a wide range of modified Reynolds numbers.

An experimental investigation of forced and free oscillations of liquid bridges positioned between two rods of equal diameter is presented. Both the resonance frequencies and damping rates for different aspect ratios of the bridge are reported. The damping rate data of the liquid bridges are obtained by high-speed videography and are the first ever reported. Resonance frequencies for the three modified Reynolds numbers of 14, 295 and 1654, and damping rates for the two modified Reynolds numbers of 14 and 295 are reported. These values of modified Reynolds numbers are generated by using ethylene glycol, distilled water, and mercury in small bridges. Gravitational effects are kept small by reducing the size of the capillary bridge. The internal flow fields of several bridges for different modified Reynolds numbers are described based on high-speed visualization. Experimental results show good agreement with results of linear and nonlinear theory.

The effect of an unsteady boundary layer on the pressure field around a bluff body has been investigated. It is found that in an unsteady flow the friction drag is always accompanied by a form drag whose magnitude is comparable with that of the former, and thus the pressure field around the unsteady boundary layer can be very different from that of an inviscid irrotational flow. The definition of the displacement thickness is modified accordingly and interpreted as a measure of the momentum of fluid trapped in the boundary layer rather than as the distance displaced laterally by the retardation of the flow in it. The result is consistent with previous specific numerical and analytical descriptions of these boundary-layer flows.

In this investigation we explore the effect of unsteady vortical structures on the adiabatic wall temperature distribution in an impinging jet. Treating first the simpler case of a free jet, we introduce a conceptual model for the separation of the total temperature, appealing to the dynamics of particle pathlines and vortex rings in the jet. The presence of a region of higher total temperature on the inside of the jet and a region of lower total temperature toward the jet periphery, predicted by the model, exhibits good agreement with the experimental data taken at high subsonic Mach number. The results from a numerical simulation further confirm the theoretical expectations.

Through a similar argument, we show that when a thermally insulated flat plate is inserted into the jet, the wall temperature distribution is modified by the presence of secondary vortical structures, which are induced near, and swept over, the plate surface. When the plate is near the jet nozzle, a region of lower wall temperature, attributable to these additional vortices, is observed in the experimental data. When the plate is further from the nozzle, no secondary vortices are formed and no region of lowered wall temperature is measured. Self-sustaining acoustic resonance, when it occurs, is found to alter significantly this picture of the wall temperature distribution.

Although the scope of this work is limited to free and impinging jets, this present topic, along with the previously reported mechanism of the Eckert–Weise effect, exemplifies the wider family of problems in which unsteady vortical structure strongly affects the wall temperature and heat transfer.