Natural convection in horizontally heated spherical fluid-filled cavities is considered in the low Grashof number limit. The equations governing the asymptotic expansion are derived for all orders. At each order a Stokes problem must be solved for the momentum correction. The general solution of the Stokes problem in a sphere with arbitrary smooth body force is derived and used to obtain the zeroth-order (creeping) flow and the first-order corrections due to inertia and buoyancy. The solutions illustrate the two mechanisms adduced by Mallinson & de Vahl Davis (1973, 1977) for spanwise flow in horizontally heated cuboids. Further, the analytical derivations and expressions clarify these mechanisms and the conditions under which they do not operate. The inertia and buoyancy effects vanish with the Grashof and Rayleigh numbers, respectively, and both vanish if the flow is purely vertical, as in a very tall and narrow cuboid.

We develop an efficient high-order boundary-element method with the mixed-Eulerian–Lagrangian approach for the simulation of fully nonlinear three-dimensional wave–wave and wave–body interactions. For illustration, we apply this method to the study of two three-dimensional steep wave problems. (The application to wave–body interactions is addressed in an accompanying paper: Liu, Xue & Yue 2001.) In the first problem, we investigate the dynamics of three-dimensional overturning breaking waves. We obtain detailed kinematics and full quantification of three-dimensional effects upon wave plunging. Systematic simulations show that, compared to two-dimensional waves, three-dimensional waves generally break at higher surface elevations and greater maximum longitudinal accelerations, but with smaller tip velocities and less arched front faces. For the second problem, we study the generation mechanism of steep crescent waves. We show that the development of such waves is a result of three-dimensional (class II) Stokes wave instability. Starting with two-dimensional Stokes waves with small three-dimensional perturbations, we obtain direct simulations of the evolution of both L2 and L3 crescent waves. Our results compare quantitatively well with experimental measurements for all the distinct features and geometric properties of such waves.

The mixed-Eulerian–Lagrangian method using high-order boundary elements, described in Xue et al. (2001) for the simulation of fully nonlinear three-dimensional wave–wave and wave–body interactions, is here extended and applied to the study of two nonlinear three-dimensional wave–body problems: (a) the development of bow waves on an advancing ship; and (b) the steep wave diffraction and nonlinear high-harmonic loads on a surface-piercing vertical cylinder. For (a), we obtain convergent steady-state bow wave profiles for a flared wedge, and the Wigley and Series 60 hulls. We compare our predictions with experimental measurements and find good agreement. It is shown that upstream influence, typically not accounted for in quasi-two-dimensional theory, plays an important role in bow wave prediction even for fine bows. For (b), the primary interest is in the higher-harmonic ‘ringing’ excitations observed and quantified in experiments. From simulations, we obtain fully nonlinear steady-state force histories on the cylinder in incident Stokes waves. Fourier analysis of such histories provides accurate predictions of harmonic loads for which excellent comparisons to experiments are obtained even at third order. This confirms that ‘ringing’ excitations are directly a result of nonlinear wave diffraction.

In this paper we study the lift-off to equilibrium of a single circular particle in Newtonian and viscoelastic fluids by direct numerical simulation. A particle heavier than the fluid is driven forward on the bottom of a channel by a plane Poiseuille flow. After a certain critical Reynolds number, the particle rises from the wall to an equilibrium height at which the buoyant weight just balances the upward thrust from the hydrodynamic force. The aim of the calculation is the determination of the critical lift-off condition and the evolution of the height, velocity and angular velocity of the particle as a function of the pressure gradient and material and geometric parameters. The critical Reynolds number for lift-off is found to be larger for a heavier particle whereas it is lower for a particle in a viscoelastic fluid. A correlation for the critical shear Reynolds number for lift-off is obtained. The equilibrium height increases with the Reynolds number, the fluid elasticity and the slip angular velocity of the particle. Simulations of single particle lift-off at higher Reynolds numbers in a Newtonian fluid by Choi & Joseph (2001) but reported here show multiple steady states and hysteresis loops. This is shown here to be due to the presence of two turning points of the equilibrium solution.

We study the transport of a slurry of heavier-than-liquid circular particles in a plane pressure-driven flow in a direct simulation. The flow is calculated in a periodic domain containing 300 circular particles. The study leads to the concept of fluidization by lift in which all the particles are suspended by lift forces against gravity perpendicular to the flow. The study is framed as an initial-value problem in which a closely packed cubic array of particles resting on the bottom of the channel is lifted into suspension. All the details of the flow are resolved numerically without model assumptions. The fluidization of circular particles first involves bed inflation in which liquid is driven into the bed by high pressure at the front and low pressure at the back of each circle in the top row. This kind of bed inflation occurs even at very low Reynolds numbers but it takes more time for the bed to inflate as the Reynolds number is reduced. It appears that the bed will not inflate if the shear Reynolds number is below the critical value for single particle lift-off. The flows with a single particle are completely determined by a shear Reynolds number and a gravity parameter when the density ratio and aspect ratio parameters are specified. In the multi-particle case, the volume fraction and distribution also matters. The transition to a fully fluidized slurry by waves is discussed.

An analytical model of the steady motion of a single particle dragged forward in a Poiseuille flow is derived and compared with a simulation. The undisturbed fluid velocity is always larger than the particle velocity, producing a fluid hold-up. The effect of the hold-up in the many particle case is to greatly reduce the velocity of the mixture which may be described by a two-fluid model in which the solid laden mixture is regarded as a second fluid with effective properties.

Motivated by the phenomena of blocked and zonal flows in Earth's atmosphere, we conducted laboratory experiments and numerical simulations to study the dynamics of an eastward jet flowing over wavenumber-two topography. The laboratory experiments studied the dynamical behaviour of the flow in a barotropic rotating annulus as a function of the experimental Rossby and Ekman numbers. Two distinct flow patterns, resembling blocked and zonal flows in the atmosphere, were observed to persist for long time intervals.

Earlier model studies had suggested that the atmosphere's normally upstream- propagating Rossby waves can resonantly lock to the underlying topography, and that this topographic resonance separates zonal from blocked flows. In the annulus, the zonal flows did indeed have super-resonant mean zonal velocities, while the blocked flows appear subresonant. Low-frequency variability, periodic or irregular, was present in the measured time series of azimuthal velocity in the blocked regime, with dominant periodicities in the range of 6–25 annulus rotations. Oscillations have also been detected in zonal states, with smaller amplitude and similar frequency. In addition, over a large region of parameter space the two flow states exhibited spontaneous, intermittent transitions from the one to the other.

We numerically simulated the laboratory flow geometry in a quasi-geostrophic barotropic model over a similar range of parameters. Both flow regimes, blocked and zonal, were reproduced in the simulations, with similar spatial and temporal characteristics, including the low-frequency oscillations associated with the blocked flow. The blocked and zonal flow patterns are present over wide ranges of forcing, topographic height, and bottom friction. For a significant portion of parameter space, both model flows are stable. Depending on the initial state, either the blocked or the zonal flow is obtained and persists indefinitely, showing the existence of multiple equilibria.

It is proved that if, in addition to the opening angle of the bounding conical streamsurface and the circulation thereon, one of the radial velocity, the radial tangential stress or the pressure on the bounding streamsurface is given, then a conically self-similar free-vortex solution is uniquely determined in the entire conical domain. In addition, it is shown that for ows inside a cone the same conclusion holds for the Yih et al. (1982) parameter T, but for exterior flows it is shown numerically that non-uniqueness may occur. For given values of the opening angle of the bounding conical streamsurface and the circulation thereon the asymptotic analysis of Shtern & Hussain (1996) is applied to obtain asymptotic formulae which interrelate the opening angle of the cone along which the jet fans out and the radial tangential stress on the bounding surface. A striking property of these formulae is that the opening angle of the cone along which the jet fans out is independent of the value of the viscosity as long as it is small enough for the first-order asymptotic expressions to apply. However, these formulae are shown to be inaccurate for moderate values of the ratio of the circulation at the bounding surface and the viscosity. To amend this shortcoming, an alternative, more accurate, asymptotic analysis is developed to derive second-order correction terms, which considerably improve the accuracy.

Perturbations of channel geometry (like variations of channel curvature or channel width) in meandering rivers give rise to morphodynamic effects which display themselves through the development of large-scale perturbations of bottom topography in the form of stationary bars developing in the longitudinal direction. The latter may then drive the lateral migration of the channel by enhancing bank erosion at bar pools: through this mechanism local perturbations of channel geometry may affect the planimetric development of meandering rivers on large timescales. The problem tackled herein is whether such morphodynamic influence is invariably felt downstream as the commonly employed model of river meandering would suggest.

In order to solve this problem, we derive the exact solution of the linearized form of the mathematical problem of river morphodynamics. Linear analysis had pointed out the existence of a resonance phenomenon: in a linear (hence ideal) context, resonance occurs when the meander wavenumber and the width ratio of the channel take values (λR and βR, respectively) such as to force free spatial modes of the system consisting of free bars which neither grow nor decay either in time or in space. Channels characterized by values of the width ratio β larger (smaller) than βR are called super- (sub-)resonant. The present solution, which applies to channels with constant width and arbitrary curvature distribution, shows that two distinct scenarios may occur: downstream influence is associated with sub-resonant channels and vice versa dominant upstream influence occurs in super-resonant channels. Small-amplitude waves of bottom topography are shown to migrate downstream in the former case and may migrate upstream in the latter, as resonance also defines the threshold conditions below (above) which small-amplitude alternate bar perturbations (may) migrate downstream (upstream).

These results have several implications. In the present paper we examine the overdeepening phenomenon whereby abrupt variations of channel curvature, as in sequences of straight and constant curvature reaches, lead to sequences of stationary alternate bars with amplitude decaying in the longitudinal direction. We show that, along with downstream overdeepening, an upstream overdeepening scenario is predicted in the super-resonant regime.

Implications of the upstream influence on planimetric development of meandering rivers are investigated in Part 2.

The exact solution of the problem of river morphodynamics derived in Part 1 is employed to formulate and solve the problem of planimetric evolution of river meanders. A nonlinear integrodifferential evolution equation in intrinsic coordinates is derived. An exact periodic solution of such an equation is then obtained in terms of a modified Fourier series expansion such that the wavenumbers of the various Fourier modes are time dependent. The amplitudes of the Fourier modes and their wavenumbers satisfy a nonlinear system of coupled ordinary differential equations of the Landau type. Solutions of this system display the occurrence of two possible scenarios. In the sub-resonant regime, i.e. when the aspect ratio of the channel is smaller than the resonant value, meandering evolves according to the classical picture: a periodic train of small-amplitude sine-generated meanders migrating downstream evolve into the classical, upstream skewed, train of meanders of Kinoshita type. Evolution displays all the experimentally observed features: the meander growth rate increases up to a maximum and then decreases, while the migration speed decreases monotonically. No equilibrium solutions are found. In the super-resonant regime the picture is essentially reversed: downstream skewing develops while meanders migrate upstream.

Numerical solutions of the planimetric evolution equation are obtained for the case when the initial channel pattern exhibits random small perturbations of the straight configuration. Under these conditions, the evolution displays the typical features of solutions of the Ginzburg–Landau equation, in particular, the occurrence of spatial modulations of the meandering pattern which organizes itself in the form of wavegroups. Furthermore, multiple loops develop in the advanced stage of meander growth.

Oblique shock reflection from an axis of symmetry is studied using Whitham's theory of geometrical shock dynamics, and the results are compared with previous numerical simulations of the phenomenon by Hornung (2000). The shock shapes (for strong and weak shocks), and the location of the shock-shock (for strong shocks), are in good agreement with the numerical results, though the detail of the shock reflection structure is, of course, not resolved by shock dynamics. A guess at a mathematical form of the shock shape based on an analogy with the Guderley singularity in cylindrical shock implosion, in the form of a generalized hyperbola, fits the shock shape very well. The smooth variation of the exponent in this equation with initial shock angle from the Guderley value at zero to 0.5 at 90° supports the analogy. Finally, steady-flow shock reflection from a symmetry axis is related to the self-similar flow.

Wave-like motion in a periodic structure of bubbles that steadily moves through ideal incompressible liquid is considered. The wavelength is microscopically short. Some general local properties containing general information about two-phase flow are found. The dynamics of small-amplitude disturbances is studied in linear systems (called trains) and in spatial structures (such as a cubic lattice). The behaviour of one-dimensional waves in various structures is shown to differ widely: one-dimensional waves in the train do not magnify, whereas in the three-dimensional structure there may be stability and instability of one-dimensional waves. In the continuum limit the one-dimensional instability is demonstrated not to be related to the mean parameters of two-phase flow. The long-wave dynamics is shown to depend significantly on the relative velocity vector orientation in the lattice, but orientation is not included in the usual equations for the two-phase continuum. One result of this study is the relation between the short-wave-type instability of the periodic structure, on the one hand, and the instability of one-dimensional flow of inviscid bubbly liquid discovered by van Wijngaarden on the other. Long microscopic waves are analysed to determine the coefficients of one-dimensional equations for a two-phase continuum model. The velocity orientation at which the coefficients of the traditional one-dimensional model are obtained is found. Short waves in a stationary structure are studied by using the system of equations based on the equation of motion of a small sphere in a general potential flow. A refined equation for the force applied on a sphere in a non-uniform potential flow is derived.

The mechanisms of sound generation in a Mach 0.9, Reynolds number 3600 turbulent jet are investigated by direct numerical simulation. Details of the numerical method are briefly outlined and results are validated against an experiment at the same flow conditions (Stromberg, McLaughlin & Troutt 1980). Lighthill's theory is used to define a nominal acoustic source in the jet, and a numerical solution of Lighthill's equation is compared to the simulation to verify the computational procedures. The acoustic source is Fourier transformed in the axial coordinate and time and then filtered in order to identify and separate components capable of radiating to the far field. This procedure indicates that the peak radiating component of the source is coincident with neither the peak of the full unfiltered source nor that of the turbulent kinetic energy. The phase velocities of significant components range from approximately 5% to 50% of the ambient sound speed which calls into question the commonly made assumption that the noise sources convect at a single velocity. Space–time correlations demonstrate that the sources are not acoustically compact in the streamwise direction and that the portion of the source that radiates at angles greater than 45° is stationary. Filtering non-radiating wavenumber components of the source at single frequencies reveals that a simple modulated wave forms for the source, as might be predicted by linear stability analysis. At small angles from the jet axis the noise from these modes is highly directional, better described by an exponential than a standard Doppler factor.

A channel flow DNS database at Reτ = 590 is used to assess the validity of modelling the redistribution term in the Reynolds stress transport equations by elliptic relaxation. The model assumptions are found to be globally consistent with the data. However, the correlation function between the fluctuating velocity and the Laplacian of the pressure gradient, which enters the integral equation of the redistribution term, is shown to be anisotropic. It is elongated in the streamwise direction and strongly asymmetric in the direction normal to the wall, in contrast to the isotropic, exponential model representation used in the original elliptic relaxation model. This discrepancy is the main cause of the slight amplification of the energy redistribution in the log layer as predicted by the elliptic relaxation equation. New formulations of the model are proposed in order to correct this spurious behaviour, by accounting for the rapid variations of the length scale and the asymmetrical shape of the correlation function. These formulations do not rely on the use of so-called ‘wall echo’ correction terms to damp the redistribution. The belief that the damping is due to the wall echo effect is called into question through the present DNS analysis.

A contact line on a heated oscillating plate is investigated. The interface is a non-deformable plane and the contact angle is π/2. The amplitude of the oscillation and the temperature deviation of the plate from the ambient temperature of the fluid are assumed to be much smaller than the viscous velocity scale. This flow is then governed by the unsteady Stokes equations coupled to the heat equation in a frame of reference moving with the contact line. Evaporation is assumed to be neglible, but the effects of heat transfer across the interface and unsteadiness are assumed to be significant. For a stationary heated plate, there are two distinct regions of flow that is induced by Marangoni stresses. An outer stagnation-point-type flow is seen, which separates from the plate for non-zero Biot numbers. For an oscillatory, isothermal plate, vortices are generated at the plate during plate reversal and are propagated along the interface. Dissipation of these vortices occurs on the Stokes layer scale. The order-Péclet-number correction in the thermal field is also found, and the presence of the flow field leads to a heated region in the steady case along the separating streamline. For the unsteady case, a localized cooled region propagates into the bulk with a trajectory determined by the relative scale of the thermal diffusive scale and the rate of heat transfer across the interface.

The axisymmetry-breaking three-dimensional instability of the axisymmetric flow between a rotating lid and a stationary cylinder is analysed. The flow is governed by two parameters – the Reynolds number Re and the aspect ratio γ (=height/radius). Published experimental results indicate that in different ranges of γ axisymmetric or non-axisymmetric instabilities can be observed. Previous analyses considered only axisymmetric instability. The present analysis is devoted to the linear stability of the basic axisymmetric flow with respect to the non-axisymmetric perturbations. After the linearization the stability problem separates into a family of quasi-axisymmetric subproblems for discrete values of the azimuthal wavenumber k. The computations are done using the global Galerkin method. The stability analysis is carried out at various densely distributed values of γ in the range 1 < γ < 3.5. It is shown that the axisymmetric perturbations are dominant in the range 1.63 < γ < 2.76. Outside this range, for γ < 1.63 and for γ > 2.76, the instability is three-dimensional and sets in with k = 2 and k = 3 or 4, respectively. The azimuthal periodicity, patterns, characteristic frequencies and phase velocities of the dominant perturbations are discussed.

We consider the spin-up of a two-layer, stably (density) stratified fluid in a rotating container with an axisymmetric sloping base and cylindrical walls. Details of the spin- up readjustment mechanisms are presented under the assumption of small impulsive changes in the rotation rate of the container. It is shown that the relative positions of the density interface and the discontinuity in wall slope determine the qualitative large-time spin-up response of the fluid. The density interface leads to a spin-up readjustment in each of the fluid layers that is essentially independent. However, when the density interface is below the boundary-slope discontinuity, a sub-region of the upper layer is predicted to readjust in an algebraic rather than exponential manner. A detailed sequence of laboratory experiments have been performed to confirm the predictions of the linear spin-up analysis.