Microscale breaking wind waves cover much of the surface of open waters exposed to moderate wind forcing. Recent studies indicate that understanding the nature and key features of the surface skin flows associated with these small waves is fundamental to explaining the dramatic enhancement of constituent exchange that occurs in their presence. We describe a laboratory study in which velocity measurements were made within a few hundred micrometres of the surface of microscale breaking wind waves without bubble entrainment, using flow visualization and particle image velocimetry (PIV) techniques for a range of wind speed and fetch conditions. Our measurements show that for each experiment, the mean surface drift directly induced by the wind on the upwind faces and crests of these waves is ($0.23\,{\pm}\,0.02$)${u}^a_\ast$ in the trough increasing to ($0.33\,{\pm}\,0.07$)${u}^a_\ast$ at the crest, where ${u}^a_\ast$ is the wind friction velocity. About these mean values, there is substantial variability in the instantaneous surface velocity up to approximately ${\pm}\,0.17{u}^a_\ast$ in the trough and ${\pm}\,0.37{u}^a_\ast$ at the crest. This variability can be attributed primarily to the modulation of the wave field, with additional contributions arising from fluctuations in wind forcing and near-surface turbulence generated by shear in the drift layer or by the influence of transient microscale breaking.

An extension to the theory of detonation shock dynamics is made and new propagation laws are derived for steady, near-CJ (Chapman–Jouguet), weakly curved detonations for a chain-branching reaction model having two components. The first is a thermally neutral induction stage governed by an Arrhenius reaction with a large activation energy, which terminates at a location called the transition interface, where instantaneous conversion of fuel into an intermediate species (chain radical) occurs. The second is an exothermic main reaction layer (or chain-recombination zone) having a temperature-independent reaction rate. We make an ansatz that the shock curvature is sufficiently large to have a leading-order influence on the induction zone structure, whereupon it is shown that multi-dimensional effects must necessarily be accounted for in the main reaction layer. Only for exactly cylindrical or spherical waves can such multi-dimensional effects be omitted. A requirement that the main reaction layer structure pass smoothly through a sonic plane leads to a propagation law for the detonation front: a relationship between the detonation velocity, the shock curvature and various shock arclength derivatives of the position of the transition interface.

For exactly cylindrically or spherically expanding waves, a multi-valued detonation velocity–curvature relationship is found, similar to that found previously for a state-sensitive one-step reaction. The change in this relationship is investigated as the ratio of the length of the main reaction layer to the induction layer is changed. We also discuss the implications of chain-branching reaction kinetics for the prediction of critical detonation initiation energy based on detonation-velocity curvature laws. Finally several calculations that illustrate the important effect that arclength and transverse flow variations may have on the steady propagation of non-planar detonation fronts are presented. Such variations may be important for the propagation of cellular gaseous detonation fronts and for the axial propagation of detonations in a cylindrical stick of condensed-phase explosive. We also show that the arclength variations provide a formal mechanism for the existence of steady non-planar detonation fronts having converging sections, a possibility ruled out for simple irreversible one-step reaction mechanisms where only diverging steady waves are admissible.

We study the shape of the interface in a partially filled horizontal cylinder which is rotating about its axis. Two-dimensional steady solutions for the interface height are examined under the assumptions that the filling fraction is small, inertia may be neglected, and the fluid forms a continuous film covering the surface. Three different regimes of steady solutions have been reported in the literature, corresponding to limits in which the ratio of gravitational to viscous forces (as defined in the text) is small, moderate or large. In each case, solutions have only been described analytically in the limit that surface tension effects are negligible everywhere. We use analytical and numerical methods, include surface tension and study steady solutions in a regime when the ratio of gravitational to viscous forces is large. This solution comprises a fluid pool that sits near the bottom of the cylinder and a film that coats the sides and top of the cylinder, the thickness of which can be determined by Landau–Levich–Derjaguin type arguments. We also examine the effect of surface tension on the solutions in the limits of the ratio of gravity to viscous forces being moderate and small.

Experimental and theoretical studies of the effect of an ultrasonically absorptive coating (UAC) on hypersonic boundary-layer stability are described. A thin coating of fibrous absorbent material (felt metal) was selected as a prototype of a practical UAC. Experiments were performed in the Mach 6 wind tunnel on a $7^{\circ}$ half-angle sharp cone whose longitudinal half-surface was solid and other half-surface was covered by a porous coating. Hot-wire measurements of ‘natural’ disturbances and artificially excited wave packets were conducted on both solid and porous surfaces. Stability analysis of the UAC effect on two- and three-dimensional disturbances showed that the porous coating strongly stabilizes the second mode and marginally destabilizes the first mode. These results are in qualitative agreement with the experimental data for natural disturbances. The theoretical predictions are in good quantitative agreement with the stability measurements for artificially excited wave packets associated with the second mode. Stability calculations for the cooled wall case showed the feasibility of achieving a dramatic increase of the laminar run using a thin porous coating of random microstructure.

We consider the spatio-temporal evolution of patterns in the marginally unstable Ekman layer driven by an applied shear stress. Both the normal and tangential components of the Earth's angular velocity are included in a tangent plane approximation of the oceanic boundary layer at latitude $\lambda$. The fluid motion in a layer of finite depth as well as one of infinite depth is considered. The linear instability in the infinite depth case is known to depend on the direction of the applied stress for $\lambda \ne 90\dg$, but this dependence is weak for the stress-driven Ekman layer. By contrast, the weakly nonlinear motion exhibits for finite and infinite depths qualitatively different dynamics for different stress directions.

The problem is treated by the method of multiple scales. In the case of finite depth, this leads to the Davey–Hocking–Stewartson equation, an amplitude equation of complex Ginzburg–Landau type coupled to a Poisson equation. In the case of infinite depth, it leads to the anisotropic complex Ginzburg–Landau equation for the amplitude of the roll motion. Motions in both finite and infinite depth basins are explored by numerical simulation, and are shown to lead to chaotic dynamics for the modulation envelope in most cases. The statistics and the nature of the patterns produced in this motion are discussed.

The propagation of nonlinear fronts in a channel flow of two contiguous homogeneous fluids of different densities is considered. Each fluid layer is of finite depth. The study is restricted to steady flows in a frame of reference moving with the front. The full governing equations are integrated numerically. The numerical method is based on boundary integral equation techniques. Although the propagation of waves in two-layer fluids is a classical problem, this is the first time that fronts have been directly computed. The limiting configuration of fronts as their amplitude increases is discussed and shown to depend on whether the front is of elevation or of depression.

We investigated experimentally the dynamics of the three-dimensional secondary instability developing in the wake of a thin flat plate at moderate Reynolds numbers. The wake is formed as the two laminar boundary layers developing on each side merge at the trailing edge of the flat plate. Both the spatial and temporal evolution of the two- and three-dimensional instabilities are analysed by means of laser-induced visualizations of the deformation of the interface separating the two streams. It was found that although the wake may exhibit two distinct three-dimensional modes with different symmetry characteristics, Modes 1 and 2 (Lasheras & Meiburg 1990), the latter appears to be amplified first, thereafter dominating the evolution of the near wake. By varying the forcing frequency of the primary two-dimensional instability, we found that the wavelength of the three-dimensional mode is selected by the wavelength of the two-dimensional Kármán vortices, with a ratio ($(\l_{3D}/\l_{2D})$) of order one. In the far-wake region, both modes appear to grow and co-exist. Furthermore, by analysing the response of the wake to spanwise-periodic and impulsive perturbations applied at the trailing edge of the plate, we demonstrate that the nature of the secondary instability of the wake behind a thin flat plate is convective. In addition, both modes are shown to have comparable wavelengths and to be the result of the same instability mechanism.

Potential flows ${\bm u} = {\bm\nabla} \phi$ are solutions of the Navier–Stokes equations for viscous incompressible fluids for which the vorticity is identically zero. The viscous term $\mu \nabla^2 {\bm u} = \mu{\bm \nabla}\nabla^2\phi$ vanishes, but the viscous contribution to the stress in an incompressible fluid (Stokes 1850) does not vanish in general. Here, we show how the viscosity of a viscous fluid in potential flow away from the boundary layers enters Prandtl's boundary layer equations. Potential flow equations for viscous compressible fluids are derived for sound waves which perturb the Navier–Stokes equations linearized on a state of rest. These linearized equations support a potential flow with the novel features that the Bernoulli equation and the potential as well as the stress depend on the viscosity. The effect of viscosity is to produce decay in time of spatially periodic waves or decay and growth in space of time-periodic waves.

In all cases in which potential flows satisfy the Navier–Stokes equations, which includes all potential flows of incompressible fluids as well as potential flows in the acoustic approximation derived here, it is neither necessary nor useful to put the viscosity to zero.

We extend our approach for the exact solution of the Riemann problem in relativistic hydrodynamics to the case in which the fluid velocity has components tangential to the initial discontinuity. As in one-dimensional flows, we show here that the wave pattern produced in a Riemann problem with multi-dimensional relativistic flows can be predicted entirely by examining the initial conditions. Our method is logically very simple and allows for a numerical implementation of an exact Riemann solver which is both straightforward and computationally efficient. The simplicity of the approach is also important for revealing special relativistic effects responsible for a smooth transition from one wave pattern to another when the tangential velocities in the initial states are suitably varied. Although this paper is focused on a flat space–time, the local Lorentz invariance allows its use also in fully general relativistic calculations.

Bounds on high-order derivative moments of a passive scalar are obtained for large values of the Schmidt number, $Sc$. The procedure is based on the approach pioneered by Batchelor for the viscous–convective range. The upper bounds for derivative moments of order $n$ are shown to grow as $Sc^{n/2}$ for very large Schmidt numbers. The results are consistent with direct numerical simulations of a passive scalar, with $Sc$ from 1/4 to 64, mixed by homogeneous isotropic turbulence. Although the analysis does not provide proper bounds for normalized moments, the combination of analysis and numerical data suggests that they decay with $Sc$, at least for odd orders.

The behaviour of incompressible side-branching flows is examined theoretically at high Reynolds numbers and compared with direct numerical simulation at moderate Reynolds numbers. The theoretical model assumes the branching (daughter) tube is small compared to the main (mother) tube and that the branching angle is small. The theory is applicable to steady and unsteady flows in two or three dimensions, and to a broad range of flow splits between mother and daughter vessels. The first main result of the work is that, in the vicinity of the branch, the flow adjusts to the imposed downstream pressure in the daughter tube through a jump (a rapid change over a short length scale) in flow properties across the daughter entrance. It is shown that, for large pressure drops in the daughter tube, fluid is sucked in at high velocities from the mother and thereby provides a favourable upstream feedback. This counteracts the tendency of the flow to separate from what would otherwise be an adversely shaped upstream wall. Increased divergence of mother and daughter tubes can thus be achieved at high daughter flow rates without separation. The second main result of the work is that the direct numerical simulations confirm the very rapid variation in flow properties and show reasonable agreement with the theory at moderate Reynolds numbers.

The jetting effect often appears in the Mach reflection of a shock and in more complicated irregular shock reflections. It also occurs in some natural phenomena, and industrially important processes. It is studied numerically using a W-modification of the second-order Godunov scheme, to integrate the system of Euler equations. It is shown that there is no correspondence between the shock reflection patterns and the occurrence of jetting. Furthermore, there are two kinds of jetting: strong which occurs when there is a branch point on the ramp surface where the streamlines divide into an upstream moving jet and a downstream moving slug; and weak which has no branch point and may occur at small and large values of the ramp angle $\theta_{w}$. The width of the jet for Mach and other reflections is determined by the angle of the Mach stem at the triple point (also called the Mach node or three-shock node). Strong jetting is unstable and the primary instability is in the jet itself. The contact discontinuity is also unstable, but its instability is secondary with respect to the jet instability. Two types of irregular reflection are identified in the dual-solution-domain. They are a two-node system comprising a Mach node followed by a four-shock (overtake) node; and another which seems to be intermediate between the previous system and a three-node reflection, which was first hypothesized by Ben-Dor & Glass (1979). An approximate criterion for the jetting $\,{\leftrightarrow}\,$ no-jetting transition is presented. It is derived by an analysis of the system of Euler equations for a self-similar flow, and has a simple geometrical interpretation.

We consider the general motion (translation and rotation) of a deformable or rigid body of arbitrary shape in a linear shear flow of an effectively inviscid and incompressible fluid possessing uniform vorticity. The ambient vorticity may be time-dependent. For two-dimensional configurations a solution with uniform vorticity is possible for all times and for three-dimensional, it is possible only initially or during a short time interval after the body is impulsively introduced into the fluid. General analytic expressions for the vortical force and moment exerted on an arbitrary moving body are presented. Bearing in mind applications for large non-spherical bubble dynamics, the general expressions for the hydrodynamic loads are further reduced for symmetric quadratic shapes such as two-dimensional ellipses or three-dimensional ellipsoids. The simplified expressions are given in terms of the body's added-mass tensor, its six velocities and the ambient vorticity. The few available degenerate solutions for cylinders and spheres are readily obtained as limiting cases.

Flotation of liquid droplets on pool surfaces, in the presence of temperature differences, is studied experimentally and numerically. Coalescence or sinking of the droplet is prevented by the thermal Marangoni motion, owing to the surface tension imbalance at the pool surface. The mechanism is the same as that investigated in previous works on coalescence and wetting prevention in the presence of temperature differences. If the droplet is colder than the liquid surface, the flow is directed radially towards the drop; this radial flow field drags the ambient air under the drop, thus creating an air film and avoiding a direct contact between the droplet and the pool molecules.

The surface velocities are measured visually with a CCD camera to image the motion of tracers floating on the pool surface; the surface temperature distributions along the pool and the droplet surfaces are measured by an infrared thermocamera. The experimental results are correlated by numerical results obtained under the assumption of spherical drop and axisymmetric flow regime. Different liquids are considered and the influence of evaporation is discussed, showing a good agreement between the experiments and the numerical simulations.

Collapse and rebound of a cavitation bubble near a wall are revisited with modern experimental means. The bubble is generated by the optical breakdown of the liquid when a strong laser pulse is focused into water. Observations are made with high-speed cinematography; framing rates range between several thousand and 100 million frames per second, and the spatial resolution is in the order of a few micrometres. After formation the bubble grows to a maximum size with a radius of 1.5 mm at the pulse energy used, and in the subsequent collapse a liquid jet evolves on the side opposite the wall and penetrates through the bubble. Using a shadowgraph technique and high framing rates, the emission of shock waves, which is observed at minimum bubble size, is resolved in detail. For a range of stand-off distances between the bubble centre and the wall, a counterjet forms during rebound. The counterjet is clearly resolved to consist of cavitation micro-bubbles, and a quantitative measure of its height evolution is given. Its emergence might be caused by a shock wave, and a possible connection of the observed shock wave scenario with the counterjet formation is discussed. No counterjets are observed when the stand-off distance is less than the maximum bubble radius, and the bubble shape becomes toroidal after the jet hits the wall. The jet impact on the wall produces a pronounced splash, which moves radially outwards in the space between the bubble and the wall. The volume compression at minimum bubble size is found to depend strongly on the stand-off distance. Some of the results are compared to numerical simulations by Tong et al. (1999), and the material presented may also be useful for comparison with future numerical work.