The classical quasi-geostrophic model in an active layer with an arbitrary vertical structure is modified by adding a boundary condition at the interface with a passive (motionless) lower layer: the difference between isopycnal and interface elevations is a Lagrangian constant, so that a particle in this boundary remains there and conserves its density. The new model has the appropriate integrals of motion: in particular, a free energy quadratic and positive definite in the deviation from a state with a uniform flow, made up of the internal and ‘external’ potential energies (due to the displacement of the isopycnals and the interface) and the kinetic energy.

Eady's model of baroclinic instability is extended with the present system, i.e. including the effect of the free lower boundary. The integrals of motion give instability conditions that are both necessary and sufficient. If the geostrophic slope of the interface is such that density increases in opposite directions at the top and bottom boundaries, then the basic flow is nonlinearly stable. For very weak internal stratification (as compared with the density jump at the interface) normal modes instability is similar to that of a simpler model, with a rigid but sloping bottom. For stronger stratification, though, the deformation of the lower boundary by the perturbation field also plays an important role, as shown in the dispersion relation, the structure of growing perturbations, and the energetics of the instability. The energy of long growing perturbations is mostly internal potential, whereas short ones have an important fraction of kinetic energy and, for strong enough stratification, external potential.

A new extension of the Kolmogorov theory, for the two-point pressure–velocity correlation, is studied by LES of homogeneous turbulence with a large inertial subrange in order to capture the high Reynolds number nonlinear dynamics of the flow. Simulations of both decaying and forced anisotropic homogeneous turbulence were performed. The forcing allows the study of higher Reynolds numbers for the same number of modes compared with simulations of decaying turbulence. The forced simulations give statistically stationary turbulence, with a substantial inertial subrange, well suited to test the Kolmogorov theory for turbulence that is locally isotropic but has significant anisotropy of the total energy distribution. This has been investigated in the recent theoretical studies of Lindborg (1996) and Hill (1997) where the role of the pressure terms was given particular attention. On the surface the two somewhat different approaches taken in these two studies may seem to lead to contradictory conclusions, but are here reconciled and (numerically) shown to yield an interesting extension of the traditional Kolmogorov theory. The results from the simulations indeed show that the two-point pressure–velocity correlation closely adheres to the predicted linear relation in the inertial subrange where also the pressure-related term in the general Kolmogorov equation is shown to vanish. Also, second- and third-order structure functions are shown to exhibit the expected dependences on separation.

The dynamics of expanding domes of isothermal lava are studied by treating the lava as a viscoplastic material with the Herschel–Bulkley constitutive law. Thin-layer theory is developed for radially symmetric extrusions onto horizontal plates. This provides an evolution equation for the thickness of the fluid that can be used to model expanding isothermal lava domes. Numerical and analytical solutions are derived that explore the effects of yield stress, shear thinning and basal sliding on the dome evolution. The results are briefly compared with an experimental study. It is found that it is difficult to unravel the combined effects of shear thinning and yield stress; this may prove important to studies that attempt to infer yield stress from morphology of flowing lava.

The weak axisymmetric fountain that results from the injection of a dense fluid upwards into a large container of homogeneous fluid of lower density is studied numerically. Using a time-accurate finite volume code, the behaviour of fountains with both a uniform and a parabolic profile of the discharge velocity at the source has been investigated. The evolution of the transient fountain flow has been analysed and two distinct stages have been identified. The time series of the passage of the fountain front has been presented and the initial, temporary and final characteristic fountain heights have been determined and scaled with the Froude number at the source. At steady state, the final fountain height and the fountain width are found to be the height and horizontal length scales which provide the full parameterization of the flow in the fountain core. The vertical velocity and temperature on the symmetry axis have been scaled with the height scale and an explicit correlation is also obtained for the former. The radial distributions of both the vertical and horizontal velocities in the zone of self-similarity in the fountain core at steady state have been scaled with the two length scales and empirical correlations have been obtained.

Some new developments of explicit algebraic Reynolds stress turbulence models (EARSM) are presented. The new developments include a new near-wall treatment ensuring realizability for the individual stress components, a formulation for compressible flows, and a suggestion for a possible approximation of diffusion terms in the anisotropy transport equation. Recent developments in this area are assessed and collected into a model for both incompressible and compressible three-dimensional wall-bounded turbulent flows. This model represents a solution of the implicit ARSM equations, where the production to dissipation ratio is obtained as a solution to a nonlinear algebraic relation. Three-dimensionality is fully accounted for in the mean flow description of the stress anisotropy. The resulting EARSM has been found to be well suited to integration to the wall and all individual Reynolds stresses can be well predicted by introducing wall damping functions derived from the van Driest damping function. The platform for the model consists of the transport equations for the kinetic energy and an auxiliary quantity. The proposed model can be used with any such platform, and examples are shown for two different choices of the auxiliary quantity.

Centrifugal buoyancy forces in a container rotating about an axis aligned with gravity lead to mean flows that interact with rotating convection. A model of the interactions between these flows and the thermal instabilities that occur in weakly nonlinear rapidly rotating convection is used to estimate when, in terms of external parameters, centrifugal buoyancy has a substantial influence on thermal convective instability. The significant physical effects include the direct action of centrifugal buoyancy on the eddies themselves, the upwards advection of basic-state vertical shear by the perturbation rolls, and the alteration of the mean thermal stratification upon which eddies grow by the basic centrifugally induced circulation. It is shown that the first effect is the most important for common laboratory settings, and can lead to destabilization of the system at outer radii. Other manifestations of centrifugal buoyancy include the generation of a positive offset of the mean temperature at the centre of the cell, and a reduction of this offset by heat fluxes arising from the centrifugally modified finite-amplitude convective eddies.

Convection rolls in a rotating layer can become unstable to the Küppers–Lortz instability. When the horizontal boundaries are stress free and the Prandtl number is finite, this instability diverges in the limit where the perturbation rolls make a small angle with the original rolls. This divergence is resolved by taking full account of the resonant mode interactions that occur in this limit: it is necessary to include two roll modes and a large-scale mean flow in the perturbation. It is found that rolls of critical wavelength whose amplitude is of order ε are always unstable to rolls oriented at an angle of order ε2/5. However, these rolls are unstable to perturbations at an infinitesimal angle if the Taylor number is greater than 4π4. Unlike the Küppers–Lortz instability, this new instability at infinitesimal angles does not depend on the direction of rotation; it is driven by the flow along the axes of the rolls. It is this instability that dominates in the limit of rapid rotation. Numerical simulations confirm the analytical results and indicate that the instability is subcritical, leading to an attracting heteroclinic cycle. We show that the small-angle instability grows more rapidly than the skew-varicose instability.

A new small-disturbance model for a steady transonic flow of moist air with non-equilibrium and homogeneous condensation around a thin airfoil is presented. The model explores the nonlinear interactions among the near-sonic speed of the flow, the small thickness ratio and angle of attack of the airfoil, and the small amount of water vapour in the air. The condensation rate is calculated according to classical nucleation and droplet growth models. The asymptotic analysis gives the similarity parameters that govern the flow problem. Also, the flow field can be described by a non-homogeneous (extended) transonic small-disturbance (TSD) equation coupled with a set of four ordinary differential equations for the calculation of the condensate (or sublimate) mass fraction. An iterative numerical scheme which combines Murman & Cole's (1971) method for the solution of the TSD equation with Simpson's integration rule for the estimation of the condensate mass production is developed. The results show good agreement with available numerical simulations using the inviscid fluid flow equations. The model is used to study the effects of humidity and of energy supply from condensation on the aerodynamic performance of airfoils.

This paper discusses the application of the Ffowcs Williams/Hawkings equation to two-dimensional problems. A two-dimensional version of this equation is derived, which not only provides a very efficient way for numerical implementation, but also reveals explicitly the features of the source mechanisms and the characteristics of the far-field noise associated with two-dimensional problems. It is shown that the sources can be interpreted, similarly to those in three-dimensional spaces, as quadrupoles from turbulent flows, dipoles due to surface pressure fluctuations on the bodies in the flow and monopoles from non-vanishing normal accelerations of the body surfaces. The cylindrical spreading of the two-dimensional waves and their far-field directivity become apparent in this new version. It also explicitly brings out the functional dependence of the radiated sound on parameters such as the flow Mach number and the Doppler factor due to source motions. This dependence is shown to be quite different from those in three-dimensional problems. The two-dimensional version is numerically very efficient because the domains of the integration are reduced by one from the three-dimensional version. The quadrupole integrals are now in a planar domain and the dipole and monopole integrals are along the contours of the two-dimensional bodies. The calculations of the retarded-time interpolation of the integrands, a time-consuming but necessary step in the three-dimensional version, are completely avoided by making use of fast Fourier transform. To demonstrate the application of this, a vortex/airfoil interaction problem is discussed, which has many practical applications and involves important issues such as vortex shedding from the trailing edge.

Direct numerical simulation of the incompressible Navier-Stokes equations is used to study flows where laminar boundary-layer separation is followed by turbulent reattachment forming a closed region known as a laminar separation bubble. In the simulations a laminar boundary layer is forced to separate by the action of a suction profile applied as the upper boundary condition. The separated shear layer undergoes transition via oblique modes and Λ-vortex-induced breakdown and reattaches as turbulent flow, slowly recovering to an equilibrium turbulent boundary layer. Compared with classical experiments the computed bubbles may be classified as ‘short’, as the external potential flow is only affected in the immediate vicinity of the bubble. Near reattachment budgets of turbulence kinetic energy are dominated by turbulence events away from the wall. Characteristics of near-wall turbulence only develop several bubble lengths downstream of reattachment. Comparisons are made with two-dimensional simulations which fail to capture many of the detailed features of the full three-dimensional simulations. Stability characteristics of mean flow profiles are computed in the separated flow region for a family of velocity profiles generated using simulation data. Absolute instability is shown to require reverse flows of the order of 15–20%. The three-dimensional bubbles with turbulent reattachment have maximum reverse flows of less than 8% and it is concluded that for these bubbles the basic instability is convective in nature.

Consider an infinite two-dimensional acoustic waveguide containing a long rectangular obstacle placed symmetrically with respect to the centreline. We search for trapped modes, i.e. modes of oscillation at particular frequencies which decay down the waveguide. We provide analytic estimates for trapped mode frequencies and prove that the number of trapped modes is asymptotically proportional to the length of the obstacle.

The fast breeder reactors (FBR) BN600 (Russia) and Phenix (France) have been the subject of several experimental studies aimed at the observation of dynamo action. Though no dynamo effect has been identified, the possibility was raised for the FBR Superphenix (France) which has an electric power twice that of BN600 and five times larger than Phenix. We present the results of a series of experimental investigations on the secondary pumps of Superphenix. The helical sodium flow inside one pump corresponds to a maximum magnetic Reynolds number (Rm) of 25 in the experimental conditions (low temperature). The magnetic field was recorded in the vicinity of the pumps and no dynamo action has been identified. An estimate of the critical flow rate necessary to reach dynamo action has been found, showing that the pumps are far from producing dynamo action. The magnetic energy spectrum was also recorded and analysed. It is of the form k−11/3, suggesting the existence of a large-scale magnetic field. Following Moffatt (1978), this spectrum slope is also justified by a phenomenological approach.

A new approach to regular and chaotic fluid advection is presented that utilizes the Thurston–Nielsen classification theorem. The prototypical two-dimensional problem of stirring by a finite number of stirrers confined to a disk of fluid is considered. The theory shows that for particular ‘stirring protocols’ a significant increase in complexity of the stirred motion – known as topological chaos – occurs when three or more stirrers are present and are moved about in certain ways. In this sense prior studies of chaotic advection with at most two stirrers, that were, furthermore, usually fixed in place and simply rotated about their axes, have been ‘too simple’. We set out the basic theory without proofs and demonstrate the applicability of several topological concepts to fluid stirring. A key role is played by the representation of a given stirring protocol as a braid in a (2+1)-dimensional space–time made up of the flow plane and a time axis perpendicular to it. A simple experiment in which a viscous liquid is stirred by three stirrers has been conducted and is used to illustrate the theory.

A rigid body colliding with a layer of dust is capable of resuspending dust through two distinct mechanisms: a ballistic mechanism, where kinetic energy is transferred from the impacting body to dust particles through direct contact, and a hydrodynamic mechanism, where dust particles are resuspended by the flow disturbance generated by the body. In this paper, we study the hydrodynamic resuspension mechanism by considering the flow around a sphere moving either towards or away from a wall. Experiments were performed using a sphere translating at a constant velocity for Reynolds number, Re, in the range 300 to 3500, and at varying angles of approach and departure from a wall. A wider range of Re was investigated by releasing dense rigid spheres above the wall. The high Reynolds number flow past a steadily translating sphere is characterized by a recirculating wake region behind the sphere. When the sphere approaches the wall and stops on making contact with it, the wake vortex which is initially behind the sphere threads over the sphere's surface, generating a secondary vortex ring. The coherent structure, composed of the wake and secondary vortices, strikes the wall and pushes fluid or dust, initially adjacent to the wall, to one side. The resuspension of dust particles of diameter b which are initially at rest on the wall is governed by a particle Shields' parameter, θp, based on the sphere's impact velocity, U: θp = ρfU2/ (ρp−ρf)bg, where ρp and ρf are respectively the density of the dust particles and fluid. The resuspension criterion is a function of particle Reynolds number, Rep, based on the diameter and fall velocity of the dust particles and occurs when θp [ges ] θp,c where θp,c ≈ 3.0 for Rep [gsim ] 1, and θp,c ≈ 5.0/Re1/2p for Rep [lsim ] 1. The geometry of the region of dust resuspended by the sphere was studied as a function of the impact velocity, angle of impact and the properties of the dust particles. When the sphere impacts a thick layer of dust, the volume concentration of resuspended dust is sufficiently high to generate a particle-driven gravity current which transports the dust far from the point of impact. The dynamics of the gravity current were determined as a function of dust particle properties and size of the impacting sphere.

A sphere moving impulsively from rest away from a wall is found not to play a significant role in the resuspension of dust; however trailing vorticity generated on the surface of the sphere advects a large volume of fluid away from the wall, which may contain dust already in suspension.

The laminar boundary layer on a flat surface is made to separate by way of aspiration through an opposite boundary, causing approximately a 25% deceleration. The detached shear layer transitions to turbulence, reattaches, and evolves towards a normal turbulent boundary layer. We performed the direct numerical simulation (DNS) of this flow, and believe that a precise experimental repeat is possible. The pressure distribution and the Reynolds number based on bubble length are close to those on airfoils; numerous features are in agreement with Gaster's and other experiments and correlations. At transition a large negative surge in skin friction is seen, following weak negative values and a brief contact with zero; this could be described as a turbulent re-separation. Temperature is treated as a passive scalar, first with uniform wall temperature and then with uniform wall heat flux. The transition mechanism involves the wavering of the shear layer and then Kelvin–Helmholtz vortices, which instantly become three-dimensional without pairing, but not primary Görtler vortices. The possible dependence of the DNS solution on the residual incoming disturbances, which we keep well below 0.1%, and on the presence of a ‘hard’ opposite boundary, are discussed. We argue that this flow, unlike the many transitional flows which hinge on a convective instability, is fully specified by just three parameters: the amount of aspiration, and the streamwise and the depth Reynolds numbers (heat transfer adds the Prandtl number). This makes comparisons meaningful, and relevant to separation bubbles on airfoils in low-disturbance environments. We obtained Reynolds-averaged Navier–Stokes (RANS) results with simple turbulence models and spontaneous transition. The agreement on skin friction, displacement thickness, and pressure is rather good, which we attribute to the simple nature of ‘transition by contact’ due to flow reversal. In contrast, a surge of the heat-transfer coefficient violates the Reynolds analogy, and is greatly under-predicted by the models.