We investigate the detailed nature of the ‘mixing transition’ through which turbulence may develop in both homogeneous and stratified free shear layers. Our focus is upon the fundamental role in transition, and in particular the associated ‘mixing’ (i.e. small-scale motions which lead to an irreversible increase in the total potential energy of the flow) that is played by streamwise vortex streaks, which develop once the primary and typically two-dimensional Kelvin–Helmholtz (KH) billow saturates at finite amplitude.

Saturated KH billows are susceptible to a family of three-dimensional secondary instabilities. In homogeneous fluid, secondary stability analyses predict that the stream-wise vortex streaks originate through a ‘hyperbolic’ instability that is localized in the vorticity braids that develop between billow cores. In sufficiently strongly stratified fluid, the secondary instability mechanism is fundamentally different, and is associated with convective destabilization of the statically unstable sublayers that are created as the KH billows roll up.

We test the validity of these theoretical predictions by performing a sequence of three-dimensional direct numerical simulations of shear layer evolution, with the flow Reynolds number (defined on the basis of shear layer half-depth and half the velocity difference) Re = 750, the Prandtl number of the fluid Pr = 1, and the minimum gradient Richardson number Ri(0) varying between 0 and 0.1. These simulations quantitatively verify the predictions of our stability analysis, both as to the spanwise wavelength and the spatial localization of the streamwise vortex streaks. We track the nonlinear amplification of these secondary coherent structures, and investigate the nature of the process which actually triggers mixing. Both in stratified and unstratified shear layers, the subsequent nonlinear amplification of the initially localized streamwise vortex streaks is driven by the vertical shear in the evolving mean flow. The two-dimensional flow associated with the primary KH billow plays an essentially catalytic role. Vortex stretching causes the streamwise vortices to extend beyond their initially localized regions, and leads eventually to a streamwise-aligned collision between the streamwise vortices that are initially associated with adjacent cores.

It is through this collision of neighbouring streamwise vortex streaks that a final and violent finite-amplitude subcritical transition occurs in both stratified and unstratified shear layers, which drives the mixing process. In a stratified flow with appropriate initial characteristics, the irreversible small-scale mixing of the density which is triggered by this transition leads to the development of a third layer within the flow of relatively well-mixed fluid that is of an intermediate density, bounded by narrow regions of strong density gradient.

The two-dimensional problem of advection–dispersion associated with a non-aqueous-phase liquid (NAPL) pool is addressed using the boundary element method. The problem is appropriately posed with an inhomogeneous boundary condition taking into consideration the presence of the pool and the impermeable layer. We derive a Fredholm integral equation of the first kind for the concentration gradient along the pool location and compute the average mass transfer coefficient numerically using the boundary-element method. Numerical results are in agreement with asymptotic analytical solutions obtained for the cases of small and large Péclet number (Pex). The asymptotic solution for small Pex, which is obtained by applying a novel perturbation technique to the integral equation, is used to de-singularize the integral equation. Results predicted by this analysis are in good agreement with experimentally determined overall mass transfer coefficients.

In this paper we reconsider the fundamental problem of the centrifugally driven spreading of a thin drop of Newtonian fluid on a uniform solid substrate rotating with constant angular speed when surface-tension and moving-contact-line effects are significant. We discuss analytical solutions to a number of problems in the case of no surface tension and in the asymptotic limit of weak surface tension, as well as numerical solutions in the case of weak but finite surface tension, and compare their predictions for the evolution of the radius of the drop (prior to the onset of instability) with the experimental results of Fraysse & Homsy (1994) and Spaid & Homsy (1997). In particular, we provide a detailed analytical description of the no-surface-tension and weak-surface-tension asymptotic solutions. We demonstrate that, while the asymptotic solutions do indeed capture many of the qualitative features of the experimental results, quantitative agreement for the evolution of the radius of the drop prior to the onset of instability is possible only when weak but finite surface-tension effects are included. Furthermore, we also show that both a fixed- and a specific variable-contact-angle condition (or ‘Tanner law’) are capable of reproducing the experimental results well.

The phenomenon of longitudinal waves in shallow grain flows has been studied through laboratory experiments. The transport process of spherical particles on a metallic chute has been characterized for this purpose. The wave mode of material transport could be measured within selected combinations of flow parameters such as the angular inclination of the chute, the mean size of the grains and the mass flow rate. It has been observed that the moving particles tend to redistribute systematically in the direction of mean flow. As a result, nonlinear longitudinal waves evolve on the surface of the chute. Observations of the predominantly rolling mode of particle motion revealed significant particle dispersion away from the wavefronts. The frequency of inter-particle collisions was low in the dispersed flow regions but increased rapidly near the wavefronts to dissipate the excess kinetic energy, thus resulting in a large increase in the average volumetric solid fraction. In order to explain the appearance of discontinuities in the volumetric solid fraction, a theoretical model that preserves the overall balance of energy and allows a discontinuous periodic solution is examined here. The depth-averaged dispersed flow of the grains has been approximated by equations of motion similar to those of shallow fluid flow. The resistance to the rolling motion of the particles is expressed in terms of the hydrodynamic drag force. The theoretical model predicts the flow criterion for which the longitudinal waves would be self-sustaining.

Numerical finite-difference results of the full axisymmetric incompressible Navier–Stokes equations are presented for the problem of the slow axial motion of a disk particle in an incompressible, rotating fluid in a cylindrical container. The governing parameters are the Ekman number, E, the Rossby number, Ro, and the dimensionless height of the container, H (with respect to the diameter of the particle). The study concerns small values of E, Ro, and HE−1/2 and compares the numerical results with predictions of previous analytical (mostly approximate) studies. Special attention is focused on the drag force. First, developed (quasi-steady state) cases are considered. Excellent agreement with the exact linear (Ro = 0) solution of Ungarish & Vedensky (1995) is obtained when the computational Ro = 10−4. The effects of the nonlinear momentum advection terms are analysed and shown to be proportional to RoE−1/2. Next, the time-development for both (a) impulsive start and (b) start under a constant axial force are considered, and good qualitative agreement with previous analytical results (including the appearance of oscillations in case (b)) is indicated.

Experimental studies of hot water depressurization show that the pressure in a vessel, just after it is opened, drops much lower than the pressure of saturation but the liquid does not boil uniformly throughout the vessel. Essentially, liquid boiling begins only on the arrival of a ‘slow wave’ of rarefaction which moves at a low speed of approximately 10 m s−1 from the open end deep into the vessel.

To explain this phenomenon we suggest a model that takes into account the difference in phase velocities. Although this difference in bubble flows is only about 1 m s−1 it proves to be sufficient to cause bubble breakup. Using this model we obtained the ‘slow wave of boiling’ in a numerical experiment that was in good agreement with the physical experiment.

Solitary waves propagating horizontally in a stratified fluid are investigated. The fluid has a shallow layer with linear stratification and a deep layer with constant density. The investigation is both experimental and theoretical. Detailed measurements of the velocities induced by the waves are facilitated by particle tracking velocimetry (PTV) and particle image velocimetry (PIV). Particular attention is paid to the role of wave breaking which is observed in the experiments. Incipient breaking is found to take place for moderately large waves in the form of the generation of vortices in the leading part of the waves. The maximal induced fluid velocity close to the free surface is then about 80% of the wave speed, and the wave amplitude is about half of the depth of the stratified layer. Wave amplitude is defined as the maximal excursion of the stratified layer. The breaking increases in power with increasing wave amplitude. The magnitude of the induced fluid velocity in the large waves is found to be approximately bounded by the wave speed. The breaking introduces a broadening of the waves. In the experiments a maximal amplitude and speed of the waves are obtained. A theoretical fully nonlinear two-layer model is developed in parallel with the experiments. In this model the fluid motion is assumed to be steady in a frame of reference moving with the wave. The Brunt-Väisälä frequency is constant in the layer with linear stratification and zero in the other. A mathematical solution is obtained by means of integral equations. Experiments and theory show good agreement up to breaking. An approximately linear relationship between the wave speed and amplitude is found both in the theory and the experiments and also when wave breaking is observed in the latter. The upper bound of the fluid velocity and the broadening of the waves, observed in the experiments, are not predicted by the theory, however. There was always found to be excursion of the solitary waves into the layer with constant density, irrespective of the ratio between the depths of the layers.

A height- and flow-dependent model for turbulent viscosity is employed to explain the generation of sand waves in tidal seas. This new model resolves the problem of excitation of very long waves in sand wave formation, because it leads to damping of the long waves and gives a finite separation between the most excited mode and the zero mode. For parameters within their physically realistic ranges, a linear analysis of the resulting system yields a first excited mode whose wavelength is similar to the characteristic wavelength of sand waves observed in nature. The physical mechanism of sand wave formation as predicted by the new model is explained in detail. The dispersion relation obtained can be the starting point for a weakly nonlinear analysis of the system.

A mathematical model is formulated to describe the initiation and evolution of intense unsteady vorticity in a low Mach number (M), weakly viscous internal flow sustained by mass addition through the sidewall of a long, narrow cylinder. An O(M) axial acoustic velocity disturbance, generated by a prescribed harmonic transient endwall velocity, interacts with the basically inviscid rotational steady injected flow to generate time-dependent vorticity at the sidewall. The steady radial velocity component convects the vorticity into the flow. The axial velocity associated with the vorticity field varies across the cylinder radius and in particular has an instantaneous oscillatory spatial distribution with a characteristic wavelength O(M) smaller than the radius. Weak viscous effects cause the vorticity to diffuse on the small radial length scale as it is convected from the wall toward the axis. The magnitude of the transient vorticity field is larger by O(M−1) than that in the steady flow.

An initial-boundary-value formulation is employed to find nonlinear unsteady solutions when a pressure node exists at the downstream exit of the cylinder. The complete velocity consists of a superposition of the steady flow, an acoustic (irrotational) field and the rotational component, all of the same magnitude.

We consider the classical problem of the laminar flow of an incompressible rotating fluid above a rotating, impermeable, infinite disk. There is a well-known class of solutions to this configuration in the form of an exact axisymmetric solution to the Navier–Stokes equations. However, the radial self-similarity that leads to the ‘rotating- disk equations’ can also be used to obtain solutions that are non-axisymmetric in nature, although (in general) this requires a boundary-layer approximation. In this manner, we locate several new solution branches, which are non-axisymmetric travelling-wave states that satisfy axisymmetric boundary conditions at infinity and at the disk. These states are shown to appear as symmetry-breaking bifurcations of the well-known axisymmetric solution branches of the rotating-disk equations. Numerical results are presented, which suggest that an infinity of such travelling states exist in some parameter regimes. The numerical results are also presented in a manner that allows their application to the analogous flow in a conical geometry.

Two of the many states described are of particular interest. The first is an exact, nonlinear, non-axisymmetric, stationary state for a rotating disk in a counter-rotating fluid; this solution was first presented by Hewitt, Duck & Foster (1999) and here we provide further details. The second state corresponds to a new boundary-layer-type approximation to the Navier–Stokes equations in the form of azimuthally propagating waves in a rotating fluid above a stationary disk. This second state is a new non-axisymmetric alternative to the classical axisymmetric Bödewadt solution.

For the understanding of longshore currents along a natural beach, the effects of bottom unevenness are considered to be important, especially for the flow in the swash zone. Currents in the swash zone are strongly influenced by the bed slope because the effect of gravity overwhelms the effect of the depth change. In the present paper, we investigate these effects and focus on waves propagating from offshore over a flat ocean basin of constant depth to a beach with a sloping wavy bottom. The waves are incident at a small angle to the beach normal, and the bed slope in the alongshore direction is varied slowly. To simplify the problem, only cnoidal waves and solitary waves are considered and the bed level is varied sinusoidally in the longshore direction.

A perturbation method is applied to the two-dimensional nonlinear shallow water equation (two-dimensional NLSWE) for the wave motion in order to generate a more simplified model of wave dynamics consisting of a one-dimensional NLSWE for the direction normal to the beach and an equation for the alongshore direction. The first equation, the one-dimensional NLSWE, is solved by Carrier & Greenspan's transformation. The solution of the second one is found by extending Brocchini & Peregrine's solution for a flat beach. Two methods for the solution of the one- dimensional NLSWE are introduced in order to get a solution applicable to large-amplitude swash motions, where the amplitude is comparable to the beach length. One is the Maclaurin expansion of the solution around the moving shoreline, and the other is Riemann's representation of the solution, which exactly satisfies the one-dimensional NLSWE and the boundary conditions. After doing a consistency check by confirming that Riemann's method, a numerical solution, agrees with the exact solution for an infinitely long, sloping beach, we assumed that the Maclaurin series solution can also describe wave motion in the swash zone properly not only for this model but also for our ‘wavy’, finite beach model.

The solution obtained from the Maclaurin series is then plugged into the equation for the alongshore direction to calculate the shore currents induced by wave run-up and back-wash motions, where a ‘weakly two-dimensional solution’ is derived from geometrical considerations. The results show that since the water depth near the shoreline is comparable to the bed level fluctuations, the flow is strongly affected by the bed unevenness, leading to recognizable changes in shoreline movement and the time-averaged velocity and the mass flux of the flow in the swash zone. More specifically, the inhomogeneity of the alongshore mass flux generates offshore currents because of the continuity condition for the fluid mass.

In the literature it was recently reported that for hypersonic reactive flow the shock stand-off distance depends not only on a reaction rate parameter, but also on the density ratio between shock and body. This is confirmed in this paper by a theoretical approach which is based on the governing conservation equations. Reasonable simplifications are introduced which for the frozen and equilibrium case allow an analytical solution for the stand-off distance on spheres. The solution method is restricted to this area since only the stand-off distance at the stagnation point is of interest. The excellent agreement achieved for the frozen or non-reactive case with well-known solutions gives evidence for the correctness of the solution method. For the equilibrium case the solution obtained shows the same behaviour as a recent study which agrees with experimental results and numerical simulations.

A thin liquid mass of fixed volume spreading under the action of gravity on an inclined plane develops a fingering instability at the front. In this study we consider the motion of a viscous sheet down a pre-wetted plane with a large inclination angle. We demonstrate that the instability is a phase instability associated with the translational invariance of the system in the direction of flow and we analyse the weakly nonlinear regime of the instability by utilizing methods from dynamical systems theory. It is shown that the evolution of the fingers is governed by a Kuramoto–Sivashinsky-type partial differential equation with solution a saw-tooth pattern when the inclined plane is pre-wetted with a thin film, while the presence of a thick film suppresses fingering.