The physical basis for parameterizing topographic stress due to unresolved eddies is examined in a quasi-geostrophic barotropic model. Topographic stress parameterization is shown to represent two effects of eddies: attraction of the flow to a statistical equilibrium featuring topographically correlated mean currents, and dissipation of potential enstrophy. Performance is evaluated by comparing parameterized low-resolution models with explicit high-resolution models.

Two coupled problems are investigated: a complete description of long-wave vortex ring oscillations in an ideal incompressible fluid, and an examination of sound radiation by these oscillations in a weakly compressible fluid.

The first part of the paper relates to the problem of eigen-oscillations of a thin vortex ring (μ[Lt ]1) in an ideal incompressible fluid. The solution of the problem is obtained in the form of an asymptotic expansion in the small parameter μ. The complete set of three-dimensional eigen-oscillations and axisymmetric modes (two-dimensional oscillations) is obtained. It is shown that, unlike the vortex column oscillations which have the form of simple angular harmonics, the majority of eigen-oscillations of a thin vortex ring have a more complex form which is a combination of two harmonics in the leading approximation. This leads to dramatic changes in the efficiency of sound radiation produced by modes of the vortex ring in comparison with the corresponding modes of the vortex column.

In the second part of the paper the solution obtained is used to investigate the process of sound radiation by vortex perturbations in a weakly compressible fluid. The vortex ring eigen-oscillations are classified according to their sound radiation efficiency. It is shown that the modes with the dimensionless frequency ω≈1/2 radiate sound most efficiently. They are two isolated modes, two infinite families of Bessel modes and a set of axisymmetric modes. The frequencies of these modes are in the interval Δω=O(μ).

The results obtained are compared with known experimental data on acoustic radiation of a turbulent vortex ring. Within the limits of the theory derived an explanation of the main characteristics of sound radiation is presented.

Formation and evolution of N (-like) waves is studied without the restriction of low amplitude, namely weak nonlinearity. To this end, the classical piston problem of gasdynamics is investigated, in which the wave is radiated by a piston executing a single cycle of harmonic oscillation into an inviscid perfect gas. The method of analysis is based on the simple-wave theory up to the shock formation time, and beyond that time on the numerical calculation by a high-resolution TVD upwind scheme. The initial sinusoid-like wave profile is rapidly distorted as the wave propagates, and this leads to the formation of head and tail shocks. The main effects of strong nonlinearity may be listed as follows: (i) entropy production at shock fronts, (ii) the existence of waves reflected from shocks, (iii) an asymmetric wave profile stemming from the boundary condition at the source of the strongly nonlinear problem. As the result, the strongly nonlinear wave possesses the following remarkable distinctive features, in contrast to its counterpart in the weakly nonlinear regime. The tail shock is not formed at the tail of the wave, and the expansion wave behind the head shock has non-uniform intensity. The N (-like) wave propagates with some excess mass. Thereby a region with low density, associated with the entropy production, appears in the vicinity of the source.

The structure and spectrum of inertial waves of an incompressible viscous fluid inside a spherical shell are investigated numerically. These modes appear to be strongly featured by a web of rays which reflect on the boundaries. Kinetic energy and dissipation are indeed concentrated on thin conical sheets, the meridional cross-section of which forms the web of rays. The thickness of the rays is in general independent of the Ekman number E but a few cases show a scaling with E1/4 and statistical properties of eigenvalues indicate that high-wavenumber modes have rays of width O(E1/3). Such scalings are typical of Stewartson shear layers. It is also shown that the web of rays depends on the Ekman number and shows bifurcations as this number is decreased.

This behaviour also implies that eigenvalues do not evolve smoothly with viscosity. We infer that only the statistical distribution of eigenvalues may follow some simple rules in the asymptotic limit of zero viscosity.

The flow fields associated with Mach reflection wave configurations in steady flows are analysed, and an analytical model for predicting the wave configurations is proposed. It is found that provided the flow field is free of far-field downstream influences, the Mach stem heights are solely determined by the set-up geometry for given incoming-flow Mach numbers. It is shown that the point at which the Mach stem height equals zero is exactly at the von Neumann transition. For some parameters, the flow becomes choked before the Mach stem height approaches zero. It is suggested that the existence of a Mach reflection not only depends on the strength and the orientation of the incident shock wave, as prevails in von Neumann's three-shock theory, but also on the set-up geometry to which the Mach reflection wave configuration is attached. The parameter domain, beyond which the flow gets choked and hence a Mach reflection cannot be established, is calculated. Predictions based on the present model are found to agree well both with experimental and numerical results.

Numerical experiments are used to study the evolution of perturbed vortex tubes in a rotating environment in order to better understand the process of two-dimensionalization of unsteady rotating flows. We specifically consider non-axisymmetric perturbations to columnar vortices aligned along the axis of rotation. The basic unperturbed vortex is chosen to have a Gaussian cross-sectional vorticity distribution. The experiments cover a parameter space in which both the strength of the initial perturbation and the Rossby number are varied. The Rossby number is defined here as the ratio of the maximum amplitude of vorticity in the Gaussian vorticity profile to twice the ambient rotation rate. For small perturbations and small Rossby numbers, both cyclones and anticyclones behave similarly, relaxing rapidly back toward two-dimensional columnar vortices. For large perturbations and small Rossby numbers, a rapid instability occurs for both cyclones and anticyclones in which antiparallel vorticity is created. The tubes break up and then re-form again into columnar vortices parallel to the rotation axis (i.e. into a quasi-two-dimensional flow) through nonlinear processes. For Rossby numbers greater than 1, even small perturbations result in the complete breakdown of the anticyclonic vortex through centrifugal instability, while cyclones remain stable. For a range of Rossby numbers greater than 1, after the breakdown of the anticyclone, a new weaker anticyclone forms, with a small-scale background vorticity of spectral shape given approximately by the −5/3 energy spectral law.

The effect of an insoluble surfactant on the transient deformation and asymptotic shape of a spherical drop that is subjected to a linear shear or extensional flow at vanishing Reynolds number is studied using a numerical method. The viscosity of the drop is equal to that of the ambient fluid, and the interfacial tension is assumed to depend linearly on the local surfactant concentration. The drop deformation is affected by non-uniformities in the surface tension due to the surfactant molecules convection–diffusion. The numerical procedure combines the boundary-integral method for solving the equations of Stokes flow, and a finite-difference method for solving the unsteady convection–diffusion equation for the surfactant concentration over the evolving interface. The parametric investigations address the effect of the ratio of the vorticity to the rate of strain of the incident flow, the Péclet number expressing the ability of the surfactant to diffuse, the elasticity number expressing the sensitivity of the surface tension to variations in surfactant concentration, and the capillary number expressing the strength of the incident flow. At small and moderate capillary numbers, the effect of a surfactant in a non-axisymmetric flow is found to be similar to that in axisymmetric straining flow studied by previous authors. The accumulation of surfactant molecules at the tips of an elongated drop decreases the surface tension locally and promotes the deformation, whereas the dilution of the surfactant over the main body of the drop increases the surface tension and restrains the deformation. At large capillary numbers, the dilution of the surfactant and the rotational motion associated with the vorticity of the incident flow work synergistically to increase the critical capillary number beyond which the drop exhibits continuous elongation. The numerical results establish the regions of validity of the small-deformation theory developed by previous authors, and illustrate the influence of the surfactant on the flow kinematics and on the rheological properties of a dilute suspension. Surfactants have a stronger effect on the rheology of a suspension than on the deformation of the individual drops.

We investigate weakly two-dimensional weakly nonlinear weakly dispersive surface waves propagating in a turbulent flow over a gradually sloping bottom. The waves are shown to be governed by a turbulently damped variable-coefficient Kadomtsev–Petviashvili equation with periodic boundary conditions. Equations governing the lowest-order mean currents in both directions as well as the equation describing the lowest-order mean surface elevation are also derived. Solutions for the wave equation are found numerically using a Fourier pseudospectral technique in space and finite differencing in the time-like variable.

A quasi-potential approximation to the Navier–Stokes equation for low-viscosity fluids is developed to study pattern formation in parametric surface waves driven by a force that has two frequency components. A bicritical line separating regions of instability to either of the driving frequencies is explicitly obtained, and compared with experiments involving a frequency ratio of 1/2. The procedure for deriving standing wave amplitude equations valid near onset is outlined for an arbitrary frequency ratio following a multiscale asymptotic expansion of the quasi-potential equations. Explicit results are presented for subharmonic response to a driving force of frequency ratio 1/2, and used to study pattern selection. Even though quadratic terms are prohibited in this case, hexagonal or triangular patterns are found to be stable in a relatively large parameter region, in qualitative agreement with experimental results.

An axisymmetric film bridge collapses under its own surface tension, disconnecting at a pair of pinchoff points that straddle a satellite bubble. The free-boundary problem for the motion of the film surface and adjacent inviscid fluid has a finite-time blowup (pinchoff). This problem is solved numerically using the vortex method in a boundary-integral formulation for the dipole strength distribution on the surface. Simulation is in good agreement with available experiments. Simulation of the trajectory up to pinchoff is carried out. The self-similar behaviour observed near pinchoff shows a ‘conical-wedge’ geometry whereby both principal curvatures of the surface are simultaneously singular – lengths scale with time as t2/3. The similarity equations are written down and key solution characteristics are reported. Prior to pinchoff, the following regimes are found. Near onset of the instability, the surface evolution follows a direction dictated by the associated static minimal surface problem. Later, the motion of the mid-circumference follows a t2/3 scaling. After this scaling ‘breaks’, a one-dimensional model is adequate and explains the second scaling regime. Closer to pinchoff, strong axial motions and a folding surface render the one-dimensional approximation invalid. The evolution ultimately recovers a t2/3 scaling and reveals its self-similar structure.

We present a number of three-dimensional axisymmetric free boundary problems for two immiscible fluids, such as air and water. A level set method is used where the interface is the zero level set of a continuous function while the two fluids are solutions of the incompressible Navier–Stokes equation. We examine the rise and distortion of an initially spherical bubble into cap bubbles and toroidal bubbles. Steady solutions for gas bubbles rising in a liquid are computed, with favourable comparisons to experimental data. We also study the inviscid limit and compare our results with a boundary integral method. The problems of an air bubble bursting at a free surface and a liquid drop hitting a free surface are also computed.

An analysis is made of the small-amplitude capillary–gravity waves which occur on the interface of two fluids and which arise out of the interaction between the Mth and Nth harmonics of the fundamental mode. The method employed is that of multiple scales in both space and time and a pair of coupled nonlinear partial differential equations for the slowly varying wave amplitudes is derived. These equations describe, correct up to third order, the progression of a wavetrain and are generalizations of the nonlinear Schrödinger-type equations used by many authors to model wave propagation. The equations are solved and formal power series expansions of the corresponding wave profiles obtained. Many different wave configurations can arise, some symmetric others asymmetric. It is found that an important influence on the type of waves which can occur is whether the ratio of the interacting wave modes is greater or less than two. Finally, an examination of the stability of the waves to plane wave perturbations is carried out.

The paper presents a theory of nonlinear evolution and secondary instabilities in Marangoni (surface-tension-driven) convection in a two-layer liquid–gas system with a deformable interface, heated from below. The theory takes into account the motion and convective heat transfer both in the liquid and in the gas layers. A system of nonlinear evolution equations is derived that describes a general case of slow long-scale evolution of a short-scale hexagonal Marangoni convection pattern near the onset of convection, coupled with a long-scale deformational Marangoni instability. Two cases are considered: (i) when interfacial deformations are negligible; and (ii) when they lead to a specific secondary instability of the hexagonal convection.

In case (i), the extent of the subcritical region of the hexagonal Marangoni convection, the type of the hexagonal convection cells, selection of convection patterns – hexagons, rolls and squares – and transitions between them are studied, and the effect of convection in the gas phase is also investigated. Theoretical predictions are compared with experimental observations.

In case (ii), the interaction between the short-scale hexagonal convection and the long-scale deformational instability, when both modes of Marangoni convection are excited, is studied. It is shown that the short-scale convection suppresses the deformational instability. The latter can appear as a secondary long-scale instability of the short-scale hexagonal convection pattern. This secondary instability is shown to be either monotonic or oscillatory, the latter leading to the excitation of deformational waves, propagating along the short-scale hexagonal convection pattern and modulating its amplitude.

The macroscopic properties and structure of the flow of two immiscible fluids through Fontainebleau sandstone are studied by numerical simulation. The pore space geometry was obtained by X-ray microtomography (Kinney et al. 1993) and the numerical simulations were performed by a new lattice-gas cellular automaton method (Olson & Rothman 1995). We first validate the numerical method by showing that the drag on a cubic array of spherical drops matches theoretical predictions. As a further test, we present a comparison between computed relative permeability and experimental measurements on the same rock. We then present a study of fluid–fluid coupling; we find that it is significant, and that it appears to be reciprocal: the flux of one fluid due to forcing on the other is the same, regardless of which fluid is forced. Lastly, we characterize the complexity and organization of the flow by means of a statistical parameter, the skewness of the distribution of local velocities.

The vorticity jump across an unsteady curved shock propagating into a two-dimensional non-uniform flow is considered in detail. The exact general expression for the vorticity jump across a shock is derived from the gasdynamics equations. This general expression is then simplified by writing it entirely in terms of the Mach number of the shock MS and the local Mach number of the flow ahead of the shock MU.

The vorticity jump is very large at places where the curvature of the shock is very large, even in the case of weak shocks. Vortex sheets form behind shock-shocks (associated with kinks in the shock front).

The ratio of vorticity production by shock curvature to vorticity production by baroclinic effects is O(½(γ−1)M2U), where γ is ratio of specific heats, which is very small if the flow ahead of the shock is only weakly compressible. If, however, the tangential gradient along the shock of M2U is large then baroclinic production is significant; this is the case in turbulent flows with large gradients of turbulent kinetic energy ½M2U. The vorticity jump across a weak shock decreases in proportion to shock intensity if the flow ahead of the shock is rotational, rather than in proportion to the cube of shock intensity as is often assumed, and thus is not negligible. It is also shown that vorticity may be generated across a straight shock even if the flow ahead of the shock is irrotational. The importance of the contribution to the vorticity jump by non-uniformities in the flow ahead of the shock has not been recognized in the past.

Examples are given of the vorticity jump across strong and weak shocks in a variety of flows exhibiting some properties of turbulence.

We describe a theory for the removal of volatile organic chemicals from an unsaturated soil stratum consisting of highly porous coarse sand layers sandwiching a thin and semipervious lens. Each soil layer is modelled as a periodic array of spherical aggregates formed by solid grains and immobile water trapped by surface tension. Volatile chemicals are vaporized in the mobile air in pores between aggregates, dissolved in the intra-aggregate water, and adsorbed on the surface of soil grains. Using the effective transport equations derived for the aggregated soils, we consider shallow layers with sharp contrast in physical properties. An asymptotic analysis is developed for an axisymmetric geometry, yielding quasi-one-dimensional governing equations for individual layers. At the leading order the flow and the vapour transport are horizontal in the coarse layers but vertical in the semipervious lens. Numerical results are presented for a simple example to demonstrate the significance of the lens permeability, diffusivity and retardation factor, and the aggregate diffusivity in the coarse layers, on the vapour transport during the stages of contamination and air-venting.