A theory on weakly rarefied low-Mach-number flows with surface reactions based on small sticking coefficients was formulated for a binary gas mixture with an irreversible surface reaction, and then extended to a multicomponent mixture with multi-step surface reactions for the situation when all chemically active species are small in concentration compared to a major inert species. Particular interest was placed on the interaction between the Knudsen layer and the surface reactions. Results show that the Knudsen layer modifies not only the incident flux of the molecules striking the surface but also the temperature-sensitive sticking coefficients, and consequently the surface reaction rates. The surface reactions in turn modify the flow structure in the Knudsen layer through the non-zero net flux at the surface. The rate expressions for the surface reactions based on sticking coefficients were derived, and the slip boundary conditions for the temperature and the species concentration suitable for application were established. The widely used Motz–Wise correction formula for the surface reaction rate was revised and the underlying assumptions leading to its derivation were shown to be inappropriate.

The stability of a hard-core fluid jet is investigated under the assumption of small electrical conductivity of the fluid for parallel and antiparallel current systems. For axisymmetric deformations the parallel current has a destabilizing influence whereas the antiparallel current improves the stability of the configuration. However, for non-axisymmetric deformations characterized by m = 1, it is shown that the jet becomes unstable in general, both for parallel and antiparallel current systems. The presence of an external conductor has been shown to have a stabilizing influence on kink instability.

Wentzel–Kramers–Brillouin theory was employed by Grimshaw (Geophys. Fluid Dyn., vol. 6, 1974, pp. 131–148) and Achatz et al. (J. Fluid Mech., vol. 210, 2010, pp. 120–147) to derive modulation equations for non-hydrostatic internal gravity wave packets in the atmosphere. This theory allows for wave packet envelopes with vertical extent comparable to the pressure scale height and for large wave amplitudes with wave-induced mean-flow speeds comparable to the local fluctuation velocities. Two classes of exact travelling wave solutions to these nonlinear modulation equations are derived here. The first class involves horizontally propagating wave packets superimposed over rather general background states. In a co-moving frame of reference, examples from this class have a structure akin to stationary mountain lee waves. Numerical simulations corroborate the existence of nearby travelling wave solutions under the pseudo-incompressible model and reveal better than expected convergence with respect to the asymptotic expansion parameter. Travelling wave solutions of the second class also feature a vertical component of their group velocity but exist under isothermal background stratification only. These waves include an interesting nonlinear wave–mean-flow interaction process: a horizontally periodic wave packet propagates vertically while draining energy from the mean wind aloft. In the process it decelerates the lower-level wind. It is shown that the modulation equations apply equally to hydrostatic waves in the limit of large horizontal wavelengths. Aside from these results of direct physical interest, the new nonlinear travelling wave solutions provide a firm basis for subsequent studies of nonlinear internal wave instability and for the design of subtle test cases for numerical flow solvers.

We consider the problem of acoustic radiation generated by a spatial instability wave on a weakly developing shear flow. Assuming a local WKBJ approximation for the instability wave near its maximum, we compute the acoustic pressure field by using a Fourier transform along the streamwise direction. When the instability wave is close to transonic near its maximum amplitude, approximations for this pressure field are obtained by a steepest descent method. A branch cut and several saddle points are shown possibly to contribute to the approximation. A detailed analysis of these contributions is provided. The modifications of the acoustic field when we pass from subsonic to supersonic are examined. In particular, the superdirective character of the acoustic field of subsonic instability waves and the directivity pattern of supersonic waves are shown to be both compatible with our mathematical description and associated with a single saddle-point contribution.

The acoustic near field is also shown to possess a caustic around which a specific approximation is derived. In a large region of the physical space, the near field is composed of two saddle-point contributions. Close to the shear flow, one of these contributions degenerates into a branch-point contribution which always becomes dominant over the instability wave downstream of a location that is computed. An interesting phenomenon is observed in certain regions downstream of the maximum: the transverse behaviour of the instability wave has to be exponentially growing far from the shear layer to match the acoustic field. We demonstrate that this phenomenon neither requires a branch-point contribution nor a supersonic instability wave.

A stratified body of fluid with compensating horizontal temperature and salinity gradients can undergo an interleaving instability which takes the form of almost horizontal intrusions. As the amplitude of these intrusions grows they can undergo secondary instabilities which eventually leads to the mixing of the fluid in the interior of the intrusions. A previous study of the secondary instabilities focused on two-dimensional disturbances. These corresponded to experimental observations of that time which all seemed to indicate that flows were indeed two-dimensional. Some more recent experiments have shown that the initial secondary instability can make the flow three-dimensional, with the secondary instabilities taking the form of rolls with their axes aligned with the direction of the flow in the intrusions. Here we present a three dimensional stability analysis of steady finite-amplitude intrusions and look at the circumstances which can lead to the three-dimensional instabilities being more likely to be observed.

We investigate the excitation and radiation of near-inertial internal gravity waves continuously excited by a latitudinally confined temporally fluctuating wind in a numerical model of a stratified ocean on a $\unicode[STIX]{x1D6FD}$-plane at mid-latitude. The surface wind forcing contains both high- and low-frequency components which excite propagating waves and a baroclinically unstable zonal jet respectively. Wentzel–Kramers–Brillouin (WKB) ray theory implies that near-inertial waves propagate strictly towards the equator. We seek to refine this view here by (i) adding the non-traditional Coriolis force (accounting for the horizontal component of the Earth’s rotation) into the equations of motion, in order to allow poleward sub-inertial propagation to occur, and (ii) relaxing the conceptual constraint of no zonal variability, to allow the zonal jet to undergo instability, to meander and to sustain an active field of mesoscale eddies, potentially impacting the excitation of near-inertial waves. The key results are that, while (i) permits weakly stratified waveguides with sub-inertial poleward wave propagation to develop in accord with theory, the sub-inertial energy flux observed is very small compared with the equatorward flux. Thus, in terms of energy radiated from the storm track, non-traditional effects are small for wind-driven near-inertial waves. The consequences of (ii) are much more pronounced. Refinement (ii) produces a radiating wave field that is bidirectional, i.e. with both poleward and equatorward components. We show that the presence of regions of significant background vorticity with horizontal scales significantly smaller than the width of the storm track provides the scale selection mechanism to excite waves with sufficiently super-inertial frequencies to propagate poleward distances of the order of 1000 km.

In the idealized problem of homogeneous isotropic stationary inertial-range turbulence the rate of relative dispersion of an ensemble of tracer pairs can be characterized by a constant C0. In order to compute this constant with random-flight equations, however, it is necessary first to know the values of two other constants, C1 and C2, that occur in the two-particle velocity-component relations of Lagrangian tracers (Faller 1992).

C1 and C2 are found by an elaborate trial and error procedure in a new two-tracer random-flight model of dispersion that matches input and output values of these two variates. The constant C0 is then computed using the Lagrangian relations and is found to be significantly smaller than when the Eulerian Kármán/Howarth correlations are used.

The probability density distribution of tracer separations has a kurtosis slightly larger than that of a comparable Gaussian distribution. At small spacings the frequency of tracer spacings is six to ten times larger than would be expected from a Gaussian distribution. The distribution function for the speed of separation of the Lagrangian tracers has a negative skewness similar to that found for two-point Eulerian velocities.

Turbulent buoyant plumes in cold fresh water are analysed, assuming a quadratic dependence of density on temperature. The model is based on the assumption that entrainment velocity is proportional to vertical velocity in the plume. Numerical and asymptotic solutions are obtained for both rising and descending plumes from virtual sources with all possible combinations of buoyancy, volume and momentum fluxes. Physical sources can be identified as points on trajectories of plumes from virtual sources.

The zero-buoyancy condition, at which the plume and the ambient have equal densities but their temperatures are on opposite sides of the temperature of maximum density, is of particular importance. If an upwardly buoyant plume rising through a body of water reaches the surface before passing through its zero-buoyancy level, it will form a surface gravity current; otherwise, the plume water will return to the source as a fountain. The height at which zero buoyancy is attained generally decreases as the source momentum flux increases: greater plume velocity produces greater entrainment and hence more rapid temperature change. Descending plumes, if ejected downwards against upward buoyancy, may be classified as strongly or weakly forced according to whether they reach the zero-buoyancy condition before being brought to rest. If they do, they continue to descend with favourable buoyancy; otherwise, they may form an inverted fountain. Once a descending plume has attained downward buoyancy, it can continue to descend indefinitely, ultimately behaving like a plume in a fluid with a linear equation of state. In contrast, a rising plume will eventually come to rest, however large its initial upward buoyancy and momentum fluxes are.

A simple idealized model is proposed to predict the appearance of alternate bottom forms in tidal channels. The model is based on the linear stability analysis of the flat seabed driven by tidal currents. The hydrodynamics is described by means of the full three-dimensional continuity and Reynolds equations. To quantify turbulent stresses, the Boussinesq assumption is introduced and an algebraic model is used for the eddy viscosity. The morphodynamics is described by the Exner equation and a simple sediment transport predictor. When applied to tidal channels, the model predicts the appearance of alternate bottom forms if the channel width is larger than a critical value. This finding agrees with previous analyses. However, the results obtained show the existence of two different modes. Close to the conditions of incipient sediment motion or when the suspended load is intense, the model suggests the appearance of an alternate sequence of shoals and pools (the first mode), characterized by a wavelength which might be comparable with the horizontal excursion of the tide. However, under such circumstances, to provide accurate quantitative results, the model should be extended to include the effects of the local acceleration and of the possible variations of the depth and width of the channel, which are neglected in the analysis. In all the other conditions, the model predicts the appearance of a second mode, presently termed tidal alternate bars. This mode is geometrically similar to the first mode, i.e. it is characterized by depositional and erosional areas which are found in an alternate arrangement, but it has significantly shorter wavelengths. In this case, the wavelength of the bedforms scales with the water depth. The physical mechanism generating tidal alternate bars appears to be the same as that generating tidal dunes, and it cannot be described by means of a depth-averaged approach.

We investigate the effect of boundary roughness on the dynamical properties of the flow in laminar Ekman boundary layers. The study considers wavy boundaries having both horizontal wavelength and vertical extent comparable in size with the boundary layer width. In the case of flat boundaries, Ekman layers are known to be active, i.e. to affect significantly the dynamics of the mainstream flow. We show how the layer modelling needs to be modified to account for such wavy boundaries. In particular, nonlinear terms enter the laminar description. This model can be linearized in the limit of small Reynolds numbers. The resulting equations are studied using both asymptotic expansions and full numerical simulations. We find that small-scale roughness significantly alters energy dissipation in the boundary layer. This can result in either a reduction or an increase of dissipation, depending on, in particular, the orientation of the mainstream flow with respect to boundary modulation. Agreement is obtained between theoretical and computational results.

A linearized analysis is made of the canonical problem of sound production during the convection of a line vortex of strength $\Gamma$ in steady flow of water over a ‘fence’ of height $h$ on a flat wall in the presence of a vacuous cavity in the wake of the fence. The cavity is assumed to extend sufficiently far downstream for sound waves to be regarded as launched above a non-compact, pressure-release surface. Additional vorticity is released from the tip of the fence in accordance with the Kutta condition, and is convected at the mean stream speed $U$ along the free streamline boundary of the cavity. Sound pressures of opposite phases are generated by the incident and the shed vorticity. The predicted radiation consists of a pressure pulse of amplitude proportional to $\rho_oU\Gamma/h$ ($\rho_o$ being the mean water density) and width approximately equal to fence height/mean flow Mach number, produced as the vortex passes the tip of the fence. The acoustic amplitude decreases rapidly at later times because of destructive interference between the sound generated by the impinging vortex and the shed vorticity.

A theory is described for propagation of vortical waves across alternate rigid and compliant panels. The structure in the fluid side at the junction of panels is a highly vortical narrow viscous structure which is idealized as a wave driver. The wave driver is modelled as a ‘half source cum half sink’. The incoming wave terminates into this structure and the outgoing wave emanates from it. The model is described by half Fourier–Laplace transforms respectively for the upstream and downstream sides of the junction. The cases below cutoff and above cutoff frequencies are studied. The theory completely reproduces the direct numerical simulation results of Davies & Carpenter (J. Fluid Mech., vol. 335, 1997, p. 361). Particularly, the jumps across the junction in the kinetic energy integral, the vorticity integral and other related quantities as obtained in the work of Davies & Carpenter are completely reproduced. Also, some important new concepts emerge, notable amongst which is the concept of the pseudo group velocity.

The scaled vorticity Ω/N and strain ∇ ζ associated with internal waves in a weak density gradient of arbitrary depth dependence together comprise a quantity that is conserved in the usual linearized approximation. This quantity I is the volume integral of the dimensionless density DI = ½[Ω2/N2 + (∇ ζ)2]. For progressive waves the ‘kinetic’ and ‘potential’ parts are equal, and in the short-wavelength limit the density DI and flux FI are related by the ordinary group velocity: FI = DIcg. The properties of DI suggest that it may be a useful measure of local internal-wave saturation.

This work addresses the transition from 2D steady to 2D unsteady laminar flow for a fully developed regime in a symmetric wavy channel geometry. We investigate the existence and characteristics of the spatio-temporal structure of the fully developed unsteady laminar flow for those particular geometries for which the steady flow presents a periodic variation of the main stream velocity component. We perform a 2D global linear stability analysis of the fully developed steady laminar flow, and we show that, for all the geometries studied, the transition is triggered by a Hopf bifurcation associated with the breaking of the symmetries and the invariance of the steady flow. Critical Reynolds numbers, most unstable modes and their characteristics are presented for large ranges of the geometric parameters, namely wavenumber ${\it\alpha}$ from 0.3 to 5 and amplitude from 0 (straight channel) to 0.5. We show that it is possible to define geometries for which the wavenumber is proportional to the most unstable mode wavenumber for the critical Reynolds number. From this modal study we address a weakly nonlinear stability analysis with a view to obtaining the Landau coefficient $g$, and then the sub- or supercritical nature of the first bifurcation characterising the transition. We show that a critical geometric amplitude beyond which the first bifurcation is supercritical is associated with each geometric wavenumber.

The stability of uniform straining flow in a semi-infinite body of viscous fluid subjected to surface cooling is examined. The viscosity of the fluid is assumed to be a prescribed function of temperature. If the viscosity variations caused by the cooling are sufficiently large the straining flow is linearly unstable to a mode in which the rate of extension of the viscous thermal boundary layer becomes localized. The parameters of the problem are the viscosity contrast in the fluid and a dimensionless measure of the rate of strain relative to the rate of cooling. The conditions under which instability occurs are determined and the physical mechanisms responsible are examined. The results are applied to discuss the formation of some surface features in lava flows.

The reflection and transmission of a gravity wave propagating through a jet-type background flow is studied. Only the linear, non-dissipative case is treated, and the hydrostatic approximation used in a stratified non-rotating medium. The behaviour of the gravity wave in the presence of two or one critical levels is investigated. In the first case, i.e. two critical levels, it is found that for high values of the Richardson number the wave is highly attenuated. For sufficiently low values of the Richardson number overreflection and overtransmission occur. It is demonstrated that a wave generated below the jet and propagating upward takes energy from the mean flow at the upper critical level for all values of the Richardson number. The single critical level has been studied as a limiting case of two merging critical levels. In this approach it is found that the wave is not transmitted and no overreflection can occur.

We develop a theoretical analysis of the displacement of inviscid fluid particles and material surfaces caused by the unsteady flow around a solid body that is moving away from a wall. The body starts at position hs from the wall, and the material surface is initially parallel to the wall and at distance hL from it. A volume of fluid Df+ is displaced away from the wall and a volume Df- towards the wall. Df+ and Df- are found to be sensitive to the ratio hL/hs. The results of our specific calculations for a sphere can be extended in general to other shapes of bodies.

When the sphere moves perpendicular to the wall the fluid displacement and drift volume Df+ are calculated numerically by computing the flow around the sphere. These numerical results are compared with analytical expressions calculated by approximating the flow around the sphere as a dipole moving away from the wall. The two methods agree well because displacement is an integrated effect of the fluid flow and the largest contribution to displacement is produced when the sphere is more than two radii away from the wall, i.e. when the dipole approximation adequately describes the flow. Analytic expressions for fluid displacement are used to calculate Df+ when the sphere moves at an acute angle α away from the wall.

In general the presence of the wall reduces the volume displaced forward and this effect is still significant when the sphere starts 100 radii from the wall. A sphere travelling perpendicular to the wall, α = 0, displaces forward a volume Df+(0) = 4πa3hL/33/2hS when the marked surface starts downstream, or behind the sphere, and displaces a volume Df+(0) ∼ 2πa3/3 forward when it is marked upstream or in front of the body. A sphere travelling at an acute angle away from the wall displaces a volume Df+(α) ∼ Df+(0) cos α forward when the surface starts downstream of the sphere. When the marked surface is initially upstream of the sphere, there are two separate regions displaced forward and a simple cosine dependence on α is not found.

These results can all be generalized to calculate material surfaces when the sphere moves at variable speed, displacements no longer being expressed in terms of time, but in relation to the distance travelled by the sphere.

In this paper we consider the axisymmetric flow of a rotating stratified fluid into a point sink. Linear analysis of the initial value problem of flow of a linearly stratified fluid into a point sink that is suddenly switched on shows that a spatially variable selective withdrawal layer is established through the outward propagation of inertial shear waves. The amplitude of these waves decays with distance from the sink; the e-folding scale of a given mode is equal to the Rossby radius of that mode. As a consequence, the flow reaches an asymptotic state, dependent on viscosity and species diffusion, in which the withdrawal-layer structure only exists for distances less than the Rossby radius based on the wave speed of the lowest mode, R1. If the Prandtl number, Pr, is large, then the withdrawal layer slowly re-forms in a time that is O(δ2iκ-1), such that it extends out much farther to a distance that is O(R1Prδ2iδ-2e) rather than O(R1).

Because there is no azimuthal pressure gradient to balance the Coriolis force associated with the radial, sinkward flow, a strong swirling flow develops. Using scaling arguments, we conclude that this swirl causes the withdrawal-layer thickness to grow like $(ft)^{\frac{1}{3}}$, such that eventually there is no withdrawal layer anywhere in the flow domain. Scaling arguments also suggest that this thickening takes place in finite-size basins.

These analyses of swirl-induced thickening and diffusive thinning can be combined to yield a classification scheme that shows how different types of flows are possible depending on the relative sizes of a parameter J, which we define as fQ(Nhv)-1, E (the Ekman number fh2v-1), and Pr.