We do a direct two-dimensional finite-elment simulation of the Navier–Stokes equations and compute the forces which turn an ellipse settling in a vertical channel of viscous fluid in a regime in which the ellipse oscillates under the action of vortex shedding. Turning this way and that is induced by large and unequal values of negative pressure at the rear separation points which are here identified with the two points on the back face where the shear stress vanishes. The main restoring mechanism which turns the broadside of the ellipse perpendicular to the fall is the high pressure at the ‘stagnation point’ on the front face, as in potential flow, which is here identified with the one point on the front face where the shear stress vanishes.

The onset of Kármán-vortex shedding is studied experimentally in the wake of different two-dimensional bluff bodies, namely an oblong cylinder, circular cylinders and plates of rectangular cross-section. Different control measures, such as wake heating, transverse body oscillations and base bleed are investigated. As the steady-periodic Kármán shedding has previously been identified as a limit-cycle, i.e. as self-excited oscillations, the experiments are interpreted in the framework of the Stuart–Landau model. The coefficients of the Stuart–Landau equation for the characteristic vortex shedding amplitude, i.e. the linear temporal growth rate, linear frequency and the Landau constant, are fully determined for the two cylinders and in part for the plate. For this purpose transients are generated by suddenly switching transverse body oscillations or base bleed on or off. The analysis of these transients by a refined method based on complex demodulation provides reliable estimates of the model coefficients and yields an experimental validation of the concept that a global instability mode grows or decays as a whole. Also, it is demonstrated that the coefficients of the Stuart–Landau equation are independent of the experimental technique used to produce the transients.

The use of a classical eddy parametrization in the analysis of continental boundary currents leads to the diffusion of momentum and relative vorticity and fails to recognize that the relevant eddies are dominated by the conservation of potential vorticity, which in turn may produce an increase in the mean relative vorticity. To illustrate this effect, we examine a non-inflected barotropic shear flow destabilized by the cross-steam variation in the bottom topògraphy of a continental slope. The finiteamplitude evolution of the waves is analysed in a simple model with a step-like bottom topography and with a piecewise-uniform potential vorticity distribution. The increase in maximum mean vorticity is computed for various values of the Rossby number and the topographic elevation, and it is suggested that a similar effect, taking into account the isopycnal topography as well as the isobaths, could maintain the large inshore shear of the Gulf Stream. Cross-shelf transport of different water ‘types’ (i.e. potential vorticity and passive tracers) are also computed and suggested to be pertinent to the more realistic oceanic problem involving baroclinic effects. The numerical calculation employs the well-known method of contour dynamics, and the Green's function appropriate for the step-like topography is derived.

Highly swept blades are now commonly used in modern aeroengines, and in this paper we solve a model problem of relevance to the understanding and prediction of noise generation by the interaction between incident vortical disturbances and such a blade. In order to include the potentially significant effects of the blade-tip region, we consider a (semi-infinite span) quarter-plane aligned at a non-zero sweep angle to supersonic mean flow, with a single harmonic gust incident on the quarterplane from upstream. The solution is completed using a novel application of the Wiener–Hopf technique, in which the usual factorisation is carried out with respect to two independent complex variables separately, and closed-form expressions for the practically important lift per unit span are derived for both the subsonic and the supersonic leading-edge regimes. The dependence of the unsteady response on the sweep angle and the gust wavenumbers is examined, and in particular we demonstrate that the magnitude of the effects of the tip region is significantly reduced by increasing the blade sweep or by considering gusts of higher frequency. It also becomes clear that the magnitude of the unsteady response can be either decreased or increased by sweeping the blade, in a way which proves highly dependent on the particular values of the flow parameters. In addition, we consider the two critical values of the gust trace speed along the leading edge which correspond to the sonic velocities in the two spanwise directions, and for which the chordwise oscillation of the unsteady lift distribution on an infinite-span blade vanishes. In these two cases, the lift per unit span (integrated over the infinite chord) is clearly singular, but we demonstrate that the effect of including the blade tip is to smooth out just one of these singularities, and replace it instead by a relatively large, but finite, value.

Steady three-dimensional convection in the form of bimodal cells in a fluid layer heated from below with rigid boundaries is studied through numerical computations for Prandtl numbers in the range 10 [lsim ] P [lsim ] 100. The stability of the steady solutions with respect to disturbances of various symmetries has been analysed. Typically, the range of stable steady bimodal convection is restricted by the transition to oscillatory bimodal convection. The oscillations preserve the spatial symmetry of the steady bimodal convection pattern in the case of high P and higher wavenumbers, but break it in the case of lower P or lower wavenumbers in the range that has been investigated. Some comparisons are made with experimental observations. The transition from bimodal to knot convection has also been studied.

When air blows over water the wind exerts a stress at the interface thereby inducing in the water a sheared turbulent drift current. We present scaling arguments showing that, if a wind suddenly starts blowing, then the sheared drift current grows in depth on a timescale that is larger than the wave period, but smaller than a timescale for wave growth. This argument suggests that the drift current can influence growth of waves of wavelength λ that travel parallel to the wind at speed c.

In narrow ‘inner’ regions either side of the interface, turbulence in the air and water flows is close to local equilibrium; whereas above and below, in ‘outer’ regions, the wave alters the turbulence through rapid distortion. The depth scale, la, of the inner region in the air flow increases with c/u*a (u*a is the unperturbed friction velocity in the wind). And so we classify the flow into different regimes according to the ratio la/λ. We show that different turbulence models are appropriate for the different flow regimes.

When (u*a + c)/UB(λ) [Lt ] 1 (UB(z) is the unperturbed wind speed) la is much smaller than λ. In this limit, asymptotic solutions are constructed for the fully coupled turbulent flows in the air and water, thereby extending previous analyses of flow over irrotational water waves. The solutions show that, as in calculations of flow over irrotational waves, the air flow is asymmetrically displaced around the wave by a non-separated sheltering effect, which tends to make the waves grow. But coupling the air flow perturbations to the turbulent flow in the water reduces the growth rate of the waves by a factor of about two. This reduction is caused by two distinct mechanisms. Firstly, wave growth is inhibited because the turbulent water flow is also asymmetrically displaced around the wave by non-separated sheltering. According to our model, this first effect is numerically small, but much larger erroneous values can be obtained if the rapid-distortion mechanism is not accounted for in the outer region of the water flow. (For example, we show that if the mixing-length model is used in the outer region all waves decay!) Secondly, non-separated sheltering in the air flow (and hence the wave growth rate) is reduced by the additional perturbations needed to satisfy the boundary condition that shear stress is continuous across the interface.

Here we study the spin-up and spin-down of a homogeneous fluid with a free surface on an experimental ‘β-plane’ and describe the important features for both cases over a range of parameters. Quantitative values are found for the velocity fields using a new image processing technique that analyses a video record of particle motion and stores the results digitally. Streamlines, pressure fields and vorticity values are found by interpolation techniques and result in a complete description of the flow characteristics. We discuss the relationship between the results of these experiments and those observed in large-scale homogeneous models of ocean circulation, e.g. Moore (1963). This study extends the work of van Heijst et al. (1990) to the case of spin-up in a rectangular container but of non-uniform depth and we note the differences to and similarities with their observations. It is related, also, to more recent results of Maas et al. (1992), who considered spin-up on a β-plane but in a tank of very different proportions to the one considered here.

Experimental observations and linear stability calculations are presented for the stability of torsional flows of viscoelastic fluids between two parallel coaxial disks, one of which is held stationary while the other is rotated at a constant angular velocity. Beyond a critical value of the dimensionless rotation rate, or Deborah number, the purely circumferential, viscometric base flow becomes unstable with respect to a nonaxisymmetric, time-dependent motion consisting of spiral vortices which travel radially outwards across the disks. Video-imaging measurements in two highly elastic polyisobutylene solutions are used to determine the radial wavelength, wavespeed and azimuthal structure of the spiral disturbance. The spatial characteristics of this purely elastic instability scale with the rotation rate and axial separation between the disks; however, the observed spiral structure of the secondary motion is a sensitive function of the fluid rheology and the aspect ratio of the finite disks.

Very near the centre of the disk the flow remains stable at all rotation rates, and the unsteady secondary motion is only observed in an annular region beyond a critical radius, denoted R*1. The spiral vortices initially increase in intensity as they propagate radially outwards across the disk; however, at larger radii they are damped and the spiral structure disappears beyond a second critical radius, R*2. This restabilization of the base viscometric flow is described quantitatively by considering a viscoelastic constitutive equation that captures the nonlinear rheology of the polymeric test fluids in steady shearing flows. A radially localized, linear stability analysis of torsional motions between infinite parallel coaxial disks for this model predicts an instability to non-axisymmetric disturbances for a finite range of radii, which depends on the Deborah number and on the rheological parameters in the model. The most dangerous instability mode varies with the Deborah number; however, at low rotation rates the steady viscometric flow is stable to all localized disturbances, at any radial position.

Experimental values for the wavespeed, wavelength and azimuthal structure of this flow instability are described well by the analysis; however, the critical radii calculated for growth of infinitesimal disturbances are smaller than the values obtained from experimental observations of secondary motions. Calculation of the time rate of change in the additional viscous energy created or dissipated by the disturbance shows that the mechanism of instability for both axisymmetric and non-axisymmetric perturbations is the same, and arises from a coupling between the kinematics of the steady curvilinear base flow and the polymeric stresses in the disturbance flow. For finitely extensible dumb-bells, the magnitude of this coupling is reduced and an additional dissipative contribution to the mechanical energy balance arises, so that the disturbance is damped at large radial positions where the mean shear rate is large.

Hysteresis experiments demonstrate that the instability is subcritical in the rotation rate, and, at long times, the initially well-defined spiral flow develops into a more complex three-dimensional aperiodic motion. Experimental observations indicate that this nonlinear evolution proceeds via a rapid splitting of the spiral vortices into vortices of approximately half the initial radial wavelength, and ultimately results in a state consisting of both inwardly and outwardly travelling spiral vortices with a range of radial wavenumbers.

Brownian systems with stiff elastic bonds of nearly constant length, such as long chain polymer molecules, behave differently when the stiff bonds are replaced by rigid bonds of exactly constant length, i.e. in statistical mechanics real stiff systems cannot be idealized by theoretical rigid ones. It is shown that a potential force can be applied to the rigidly constrained system in order to make it behave like the limit of a very stiff elastic system. A simple explicit expression for the required potential, suitable for computer simulations of the Brownian motion, is given for general constraints and also in the particular case of a trumbbell or trimer.

When dealing with systems showing a Hopf bifurcation as the first instability from a conductive state leading to travelling waves, the distinction between convective and absolute instability becomes significant. The convectively unstable regime is characterized by the fact that a homogeneous disturbance may have a positive growth rate, while a single localized perturbation cannot trigger the onset of nonlinear convection. In this paper the convective instability occurring in binary fluid mixtures for a negative separation ratio is utilized for amplifying intrinsic thermal fluctuations, which in this way become accessible to quantitative measurements. The experiments are performed in a quasi-one-dimensional convection channel which, by means of subcritical ramps, effectively prevents the reflection of the travelling waves from the sidewalls. Thus, that range of the convective instability within which linear waves can be observed is strongly enhanced. The temperature variations involved in the observed travelling-wave states are quantified by using the shadowgraph method. By resonantly stimulating the system with its linear Hopf frequency, the reflection ability and some coefficients of the amplitude equation appropriate for describing the convection features near onset can be determined. Without stimulation, travelling-wave states of very small amplitudes showing an erratic spatio-temporal behaviour occur spontaneously inside the convectively unstable regime. The temporal correlation function calculated from the measured light intensity caused by these states is compared with a theoretical expression obtained from a Ginzburg—Landau equation containing a noise term. A very good agreement is found for the amplitude if thermal noise is assumed to be the reason for these fluctuating convection rolls, thus supporting the idea that the response of the system to thermal fluctuations is observed.

The dispersive (i.e. non-Kelvin) linear wave field on the equatorial β-plane, in a single vertical mode, is fully described by a single potential φ. Long Rossby waves, which are weakly dispersive, are represented in this field. This description is free from the problem of the ‘spurious solution’ encountered when working with an evolution equation for the meridional velocity; addition of this unwanted solution represents a gauge transformation that leaves the physical fields unaltered.

The general solution of the ray equations is found, including trajectories, and the amplitudes and phase fields. This solution is asymptotically valid for either high or low frequencies. The ray paths are identical in both limits, but the phase field is not, reflecting the isotropy of Poincaré waves, in one case, and the zonal anisotropy of Rossby waves, in the other.

Two examples are studied by ray theory: meridional normal modes and wave radiation from a point source in the equator. In the first case, the exact dispersion relation is obtained. In the second one, northern and southern caustics bend towards the equator, meeting there at focal points. The full solution is the superposition of many leaves and has a structure that would be hard to find in a normal modes expansion.

A new and very general technique for simulating solid–fluid suspensions is described; its most important feature is that the computational cost scales linearly with the number of particles. The method combines Newtonian dynamics of the solid particles with a discretized Boltzmann equation for the fluid phase; the many-body hydrodynamic interactions are fully accounted for, both in the creeping-flow regime and at higher Reynolds numbers. Brownian motion of the solid particles arises spontaneously from stochastic fluctuations in the fluid stress tensor, rather than from random forces or displacements applied directly to the particles. In this paper, the theoretical foundations of the technique are laid out, illustrated by simple analytical and numerical examples; in a companion paper (Part 2), extensive numerical tests of the method, for stationary flows, time-dependent flows, and finite-Reynolds-number flows, are reported.

A new and very general technique for simulating solid–fluid suspensions has been described in a previous paper (Part 1); the most important feature of the new method is that the computational cost scales linearly with the number of particles. In this paper (Part 2), extensive numerical tests of the method are described; results are presented for creeping flows, both with and without Brownian motion, and at finite Reynolds numbers. Hydrodynamic interactions, transport coefficients, and the short-time dynamics of random dispersions of up to 1024 colloidal particles have been simulated.

We consider a semi-infinite (bounded above) plane-parallel layer of barotropic fluid in a constant gravitational field. We present a proof that flows of such a fluid cannot be time-independent in a reference frame wherein the flow's velocity field falls off asymptotically faster than the inverse of the radial distance, R. This includes all flows of finite kinetic energy as such flows must fall off faster than R−1.5. The unsteadiness is due in part to the dynamical expansion of a compressible fluid in motion; this expansion leads to a density deficit so that in the presence of gravity the flow rises buoyantly and cannot be steady in time. The non-existence of certain classes of steady uniform-fluid flows is also discussed.

We investigate streaming in a square cavity where a lateral temperature gradient interacts with a constant gravity field modulated by small harmonic oscillations of order ε. The Boussinesq equations are expanded by regular perturbation in powers of ε, and the O(ε2) equations contain Reynolds-stress-type terms that cause streaming. The resulting hierarchy of equations is solved by finite differences to investigate the O(ε1) and O(ε2) fields and their parametric dependence on the Rayleigh number Ra, Prandtl number Pr, and forcing frequency ω. It has been found that the streaming flow is quite small at small values of Ra, but becomes appreciable at high Ra and starts to influence such flow properties as the strength of the circulation and the overall heat transfer. Under suitable parametric conditions of finite frequency and moderate Pr the periodic forcing motion interacts with the instabilities associated with the O(εo) base flow leading to resonances that become stronger as Ra increases. It is argued that these resonances will have their greatest effect on streaming for Pr ≈ 1. At low frequencies the streaming flow shows marked structural changes as Ra is increased leading to an interesting change in the sign of the O(ε2) contribution to the Nusselt number. Also, as the frequency is changed the O(ε2) Nusselt number again changes sign at approximately the resonant frequency.