Buoyancy-driven flow in a narrow-gap annulus formed by two concentric horizontal cylinders is investigated numerically. The three-dimensional transient equations of fluid motion and heat transfer are solved to study multiple supercritical states occurring within annuli having impermeable endwalls, which are encountered in various applications. For the first time, three-dimensional supercritical states are shown to occur in a narrow-gap annulus and the existence of four such states is established. These four states are characterized by the orientations and directions of rotation of counter-rotating rolls that form in the upper part of the annulus owing to thermal instability, and exhibit (i) transverse rolls, (ii) transverse rolls with reversed directions of rotation, (iii) longitudinal rolls in combination with transverse rolls, and (iv) longitudinal rolls with reversed directions of rotation in combination with transverse rolls, respectively. Simulations are performed at Rayleigh numbers approaching and exceeding the critical value to gain insight into the physical processes influencing development of the secondary flow structures. The evolution of the supercritical flow fields and temperature distributions with increasing Rayleigh number and the interaction between the secondary and primary flows are thoroughly investigated. Factors influencing the number of rolls are studied for each supercritical state. Heat transfer results are presented in the form of local Nusselt number distributions and overall annulus Nusselt numbers. Two-dimensional natural convection occurring early in the transient evolution of the flow field is also examined. Results obtained for a wide range of annulus radius ratios and Rayleigh numbers are shown to be in excellent agreement with results from previous experimental and numerical studies, thereby validating the present numerical scheme.

Although predicted early in the 20th century, a single-phase vapour rarefaction shock wave has yet to be demonstrated experimentally. Results from a previous shock tube experiment appear to indicate a rarefaction shock wave. These results are discussed and their interpretation challenged. In preparation for a new shock tube experiment, a global theory is developed, utilizing a van der Waals fluid, for demonstrating a single-phase vapour rarefaction shock wave in the incident flow of the shock tube. The flow consists of four uniform regions separated by three constant-speed discontinuities: a rarefaction shock, a compression shock, and a contact surface. Entropy jumps and upstream supersonic Mach number conditions are verified for both shock waves. The conceptual van der Waals model is applied to the fluid perfluoro-tripentylamine (FC-70, C15F33N) analytically, and verified with computational simulations. The analysis predicts a small region of initial states that may be used to unequivocally demonstrate the existence of a single-phase vapour rarefaction shock wave. Simulation results in the form of representative sets of thermodynamic state data (pressure, density, Mach number, and fundamental derivative of gas dynamics) are presented.

Choi & Joseph (2001) reported a two-dimensional numerical investigation of the lift-off of 300 circular particles in plane Poiseuille flows of Newtonian fluids. We perform similar simulations. Particles heavier than the fluid are initially placed in a closely packed ordered configuration at the bottom of a periodic channel. The fluid–particle mixture is driven by an external pressure gradient. The particles are suspended or fluidized by lift forces that balance the buoyant weight perpendicular to the flow. Pressure waves corresponding to the waves at the fluid–mixture interface are observed. During the initial transient, these waves grow, resulting in bed erosion. At sufficiently large shear Reynolds numbers the particles occupy the entire channel width during the transient. The particle bed eventually settles to an equilibrium height which increases as the shear Reynolds number is increased. Heavier particles are lifted to a smaller equilibrium height at the same Reynolds number. A correlation for the lift-off of many particles is obtained from the numerical data. The correlation is used to estimate the critical shear Reynolds number for lift-off of many particles. The critical shear Reynolds number for lift-off of a single particle is found to be greater than that for many particles. The procedures used here to obtain correlations from direct simulations in two dimensions and the type of correlations that emerge should generalize to three-dimensional simulations at present underway.

The Green function used for analysing ship motions in waves is the velocity potential due to a point source pulsating and advancing at a uniform forward speed. The behaviour of this function is investigated, in particular for the case when the source is located at or close to the free surface. In the far field, the Green function is represented by a single integral along one closed dispersion curve and two open dispersion curves. The single integral along the open dispersion curves is analysed based on the asymptotic expansion of a complex error function. The singular and highly oscillatory behaviour of the Green function is captured, which shows that the Green function oscillates with indefinitely increasing amplitude and indefinitely decreasing wavelength, when a field point approaches the track of the source point at the free surface. This sheds some light on the nature of the difficulties in the numerical methods used for predicting the motion of a ship advancing in waves.

We develop a theory of nonlinear geostrophic adjustment of arbitrary localized (i.e. finite-energy) disturbances in the framework of the non-dissipative rotating shallow-water dynamics. The only assumptions made are the well-defined scale of disturbance and the smallness of the Rossby number Ro. By systematically using the multi-time-scale perturbation expansions in Rossby number it is shown that the resulting field is split in a unique way into slow and fast components evolving with characteristic time scales f−10 and (f0Ro)−1 respectively, where f0 is the Coriolis parameter. The slow component is not influenced by the fast one and remains close to the geostrophic balance. The algorithm of its initialization readily follows by construction.

The scenario of adjustment depends on the characteristic scale and/or initial relative elevation of the free surface ΔH/H0, where ΔH and H0 are typical values of the initial elevation and the mean depth, respectively. For small relative elevations (ΔH/H0 = O(Ro)) the evolution of the slow motion is governed by the well-known quasi-geostrophic potential vorticity equation for times t [les ] (f0Ro)−1. We find modifications to this equation for longer times t [les ] (f0Ro2)−1. The fast component consists mainly of linear inertia–gravity waves rapidly propagating outward from the initial disturbance.

For large relative elevations (ΔH/H0 [Gt ] Ro) the slow field is governed by the frontal geostrophic dynamics equation. The fast component in this case is a spatially localized packet of inertial oscillations coupled to the slow component of the flow. Its envelope experiences slow modulation and obeys a Schrödinger-type modulation equation describing advection and dispersion of the packet. A case of intermediate elevation is also considered.

Experiments by Kessler on bioconvection in laboratory suspensions of bacteria (Bacillus subtilis), contained in a deep chamber, reveal the development of a thin upper boundary layer of cell-rich fluid which becomes unstable, leading to the formation of falling plumes. We use the continuum description of such a suspension developed by Hillesdon et al. (1995) as the basis for a theoretical model of the boundary layer and an axisymmetric plume. If the boundary layer has dimensionless thickness λ [Lt ] 1, the plume has width λ1/2. A similarity solution is found for the plume in which the cell flux and volume flux can be matched to those in the boundary layer and in the bulk of the suspension outside both regions. The corresponding model for a two-dimensional plume fails to give a self-consistent solution.

Meso-scale structures that take the form of clusters and streamers are commonly observed in dilute gas–particle flows, such as those encountered in risers. Continuum equations for gas–particle flows, coupled with constitutive equations for particle-phase stress deduced from kinetic theory of granular materials, can capture the formation of such meso-scale structures. These structures arise as a result of an inertial instability associated with the relative motion between the gas and particle phases, and an instability due to damping of the fluctuating motion of particles by the interstitial fluid and inelastic collisions between particles. It is demonstrated that the meso-scale structures are too small, and hence too expensive, to be resolved completely in simulation of gas–particle flows in large process vessels. At the same time, failure to resolve completely the meso-scale structures in a simulation leads to grossly inaccurate estimates of inter-phase drag, production/dissipation of pseudo-thermal energy associated with particle fluctuations, the effective particle-phase pressure and the effective viscosities. It is established that coarse-grid simulation of gas–particle flows must include sub-grid models, to account for the effects of the unresolved meso-scale structures. An approach to developing a plausible sub-grid model is proposed.

The paper considers the derivation and properties of the Fanno model for nearly unidirectional turbulent flow of gas in a tube. The model is relevant to many industrial processes. Approximate solutions are derived and numerically validated for evolving flows of initially small amplitude, and these solutions reveal the prevalence of localized large-time behaviour, which is in contrast to inviscid acoustic theory. The properties of large-amplitude travelling waves are summarized, which are also surprising when compared to those of inviscid theory.

The propagation of unsteady disturbances in a slowly varying cylindrical duct carrying mean swirling flow is described. A consistent multiple-scales solution for the mean flow and disturbance is derived, and the effect of finite-impedance boundaries on the propagation of disturbances in mean swirling flow is also addressed.

Two degrees of mean swirl are considered: first the case when the swirl velocity is of the same order as the axial velocity, which is applicable to turbomachinery flow behind a rotor stage; secondly a small swirl approximation, where the swirl velocity is of the same order as the axial slope of the duct walls, which is relevant to the flow downstream of the stator in a turbofan engine duct.

The presence of mean vorticity couples the acoustic and vorticity equations and the associated eigenvalue problem is not self-adjoint as it is for irrotational mean flow. In order to obtain a secularity condition, which determines the amplitude variation along the duct, an adjoint solution for the coupled system of equations is derived. The solution breaks down at a turning point where a mode changes from cut on to cut off. Analysis in this region shows that the amplitude here is governed by a form of Airy's equation, and that the effect of swirl is to introduce a small shift in the location of the turning point. The reflection coefficient at this corrected turning point is shown to be exp (iπ/2).

The evolution of axial wavenumbers and cross-sectionally averaged amplitudes along the duct are calculated and comparisons made between the cases of zero mean swirl, small mean swirl and O(1) mean swirl. In a hard-walled duct it is found that small mean swirl only affects the phase of the amplitude, but O(1) mean swirl produces a much larger amplitude variation along the duct compared with a non-swirling mean flow. In a duct with finite-impedance walls, mean swirl has a large damping effect when the modes are co-rotating with the swirl. If the modes are counter-rotating then an upstream-propagating mode can be amplified compared to the no-swirl case, but a downstream-propagating mode remains more damped.

A series of numerical experiments is performed to investigate the breaking of obliquely incident internal waves propagating towards a bottom slope. The case of critical reflection is considered, where the angle between the wave group velocity vector and the horizontal matches the bottom slope angle. The flow evolution is found to be significantly different from the evolution observed previously in simulations of normally incident waves. The divergence of the Reynolds stress in the breaking zone causes a strong along-slope mean current, which changes the flow structure dramatically. The wave does not penetrate the current but breaks down at its upper surface as the result of a critical layer interaction. A continuously broadening mean along-slope current with an approximately constant velocity is produced. We propose a simple model of the process based on the momentum conservation law and the radiation stress concept. The model predictions are verified against the numerical results and are used to evaluate the possible strength of along-slope currents generated by this process in the ocean.

We study the stability of stratified gas–liquid flow in a horizontal rectangular channel using viscous potential flow. The analysis leads to an explicit dispersion relation in which the effects of surface tension and viscosity on the normal stress are not neglected but the effect of shear stresses is. Formulas for the growth rates, wave speeds and neutral stability curve are given in general and applied to experiments in air–water flows. The effects of surface tension are always important and determine the stability limits for the cases in which the volume fraction of gas is not too small. The stability criterion for viscous potential flow is expressed by a critical value of the relative velocity. The maximum critical value is when the viscosity ratio is equal to the density ratio; surprisingly the neutral curve for this viscous fluid is the same as the neutral curve for inviscid fluids. The maximum critical value of the velocity of all viscous fluids is given by that for inviscid fluid. For air at 20°C and liquids with density ρ = 1 g cm−3 the liquid viscosity for the critical conditions is 15 cP: the critical velocity for liquids with viscosities larger than 15 cP is only slightly smaller but the critical velocity for liquids with viscosities smaller than 15 cP, like water, can be much lower. The viscosity of the liquid has a strong effect on the growth rate. The viscous potential flow theory fits the experimental data for air and water well when the gas fraction is greater than about 70%.

In this study we have investigated the behaviour of natural convection flow in open cavities, with both homogeneous and thermally stratified ambient, using direct numerical simulation. The cavity is insulated at the top and bottom boundaries, heated from the left-hand side boundary and open at the right-hand side. A wide range of Rayleigh numbers were considered (5 × 106 to 1 × 1010) with Pr = 0.7 for all cases. It was found that the homogeneous flow is steady for all Rayleigh numbers considered, whereas the stratified flow with a high enough Rayleigh number exhibits low- and high-frequency signals of the same type as are observed for closed cavity flow. The thermal boundary layer is examined in detail and it is shown that both low- and high-frequency signals are located predominantly in the upper region of the heated plate and are associated with a reverse-S-flow formed by the boundary layer exit jet interacting with the stratified interior. The low-frequency signal is associated with standing waves in the boundary layer, whereas the high-frequency signal is associated with travelling waves. The high-frequency signal occurs initially as a harmonic of the base low-frequency signal. A corner jet with the same inlet characteristics as the natural convection boundary layer exit jet is also examined and shown to exhibit a similar bifurcation, but with the low frequency always dominant. It is suggested that the generation mechanism for the bifurcation of the natural convection flow is the same as that for the corner jet.

Large-scale particle-driven gravity currents occur in the atmosphere, often in the form of pyroclastic flows that result from explosive volcanic eruptions. The behaviour of these gravity currents is analysed here and it is shown that compressibility can be important in flow of such particle-laden gases because the presence of particles greatly reduces the density scale height, so that variations in density due to compressibility are significant over the thickness of the flow. A shallow-water model of the flow is developed, which incorporates the contribution of particles to the density and thermodynamics of the flow. Analytical similarity solutions and numerical solutions of the model equations are derived. The gas–particle mixture decompresses upon gravitational collapse and such flows have faster propagation speeds than incompressible currents of the same dimensions. Once a compressible current has spread sufficiently that its thickness is less than the density scale height it can be treated as incompressible. A simple ‘box-model’ approximation is developed to determine the effects of particle settling. The major effect is that a small amount of particle settling increases the density scale height of the particle-laden mixture and leads to a more rapid decompression of the current.

In this numerical study, we investigate natural convection in a two-dimensional square-section enclosure vibrating sinusoidally parallel to the applied temperature gradient in a zero-gravity field. The full Navier–Stokes equations are simplified with the Boussinesq approximation and solved by a finite difference method. Whereas the Prandtl number Pr is fixed to 7.1 (except for some test cases with Pr = 7.0, 6.8), the vibrational Rayleigh number Ra based on acceleration amplitude is varied from 1.0 × 104 to 1.0 × 105, and dimensionless angular frequency ω is varied from 1.0 × 100 to 1.0 × 103. In the tested range, time evolutions exhibit synchronous, 1/2-subharmonic and non-periodic responses, and flow patterns are characterized mainly by one- or two-cell structures. Flow-regime diagrams show considerable differences from results in a non-zero-mean-gravity field even at large acceleration amplitudes, and suggest that some parts of non-periodic-response regimes may be related to transitions between flow patterns. The amplitude of fluctuations in spatially averaged kinetic energy density K (equal to the difference between maximum and minimum kinetic energies over a cycle) tends to be large when fluid is stationary everywhere over some interval of time during each period, and has a peak when fluid begins to move continuously throughout one period. Such peaks are caused by impulsively started convection, and are not connected to resonant oscillations in a constant-gravity field.

Enhanced diffusion coefficients arising from the theory of periodic homogenized averaging for a passive scalar diffusing in the presence of a large-scale, fluctuating mean wind superimposed upon a small-scale, steady flow with non-trivial topology are studied. The purpose of the study is to assess how the extreme sensitivity of enhanced diffusion coefficients to small variations in large-scale flow parameters previously exhibited for steady flows in two spatial dimensions is modified by either the presence of temporal fluctuation, or the consideration of fully three-dimensional steady flow. We observe the various mixing parameters (Péclet, Strouhal and periodic Péclet numbers) and related non-dimensionalizations. We document non-monotonic Péclet number dependence in the enhanced diffusivities, and address how this behaviour is camouflaged with certain non-dimensional groups. For asymptotically large Strouhal number at fixed, bounded Péclet number, we establish that rapid wind fluctuations do not modify the steady theory, whereas for asymptotically small Strouhal number the enhanced diffusion coefficients are shown to be represented as an average over the steady geometry. The more difficult case of large Péclet number is considered numerically through the use of a conjugate gradient algorithm. We consider Péclet-number-dependent Strouhal numbers, S = QPe−(1+γ), and present numerical evidence documenting critical values of γ which distinguish the enhanced diffusivities as arising simply from steady theory (γ < −1) for which fluctuation provides no averaging, fully unsteady theory (γ ∈ (−1, 0)) with closure coefficients plagued by non-monotonic Péclet number dependence, and averaged steady theory (γ > 0). The transitional case with γ = 0 is examined in detail. Steady averaging is observed to agree well with the full simulations in this case for Q [les ] 1, but fails for larger Q. For non-sheared flow, with Q [les ] 1, weak temporal fluctuation in a large-scale wind is shown to reduce the sensitivity arising from the steady flow geometry; however, the degree of this reduction is itself strongly dependent upon the details of the imposed fluctuation. For more intense temporal fluctuation, strongly aligned orthogonal to the steady wind, time variation averages the sensitive scaling existing in the steady geometry, and the present study observes a Pe1 scaling behaviour in the enhanced diffusion coefficients at moderately large Péclet number. Finally, we conclude with the numerical documentation of sensitive scaling behaviour (similar to the two-dimensional steady case) in fully three dimensional ABC flow.

We present and analyse a model for the spherical pulsations and translational motions of a pair of interacting gas bubbles in an incompressible liquid. The model is derived rigorously in the context of potential flow theory and contains all terms up to and including fourth order in the inverse separation distance between the bubbles. We use this model to study the cases of both weak and moderate applied acoustic forcing. For weak acoustic forcing, the radial pulsations of the bubbles are weakly coupled, which allows us to obtain a nonlinear time-averaged model for the relative distance between the bubbles. The two parameters of the time-averaged model classify four different dynamical regimes of relative translational motion, two of which correspond to the attraction and repulsion of classical secondary Bjerknes theory. Also predicted is a pattern in which the bubbles exhibit stable, time-periodic translational oscillations along the line connecting their centres, and another pattern in which there is an unstable separation distance such that bubble pairs can either attract or repel each other depending on whether their initial separation distance is smaller or larger than this value. Moreover, it is shown that the full governing equations possess the dynamics predicted by the time-averaged model. We also study the case of moderate-amplitude acoustic forcing, in which the bubble pulsations are more strongly coupled to each other and bubble translation also affects the radial pulsations. Here, radial harmonics and nonlinear phase shifting play a significant role, as bubble pairs near resonances are observed to translate in patterns opposite to those predicted by classical secondary Bjerknes theory. In this work, dynamical systems techniques and the method of averaging are the primary mathematical methods that are employed.