The most widely used definitions of a vortex are not objective: they identify different structures as vortices in frames that rotate relative to each other. Yet a frame-independent vortex definition is essential for rotating flows and for flows with interacting vortices. Here we define a vortex as a set of fluid trajectories along which the strain acceleration tensor is indefinite over directions of zero strain. Physically, this objective criterion identifies vortices as material tubes in which material elements do not align with directions suggested by the strain eigenvectors. We show using examples how this vortex criterion outperforms earlier frame-dependent criteria. As a side result, we also obtain an objective criterion for hyperbolic Lagrangian structures.

We have investigated experimentally the swirling flow between a stationary and a rotating disk with fixed closed end. In order to perform velocimetry measurements we implemented a heterodyne photon-correlation setup and obtained the three components of the velocity field at different positions along the gap between the disks. We compared the results for two different Reynolds numbers with a numerical solution of the similarity equation, to investigate the relation between the finite and infinite disk solution, theoretically studied by Brady & Durlofsky (1987). For the measurements performed at $\hbox{\it Re}$ below the critical Reynolds number $\hbox{\it Re}_C\,{=}\,80$, we found that the two solutions agree very well near the axis of rotation. Above $\hbox{\it Re}_C$ we found that this quantitative agreement no longer holds, but the flow qualitatively resembles the Batchelor solution, with two boundary layers and a core rotating as a solid body. Our results validate the theoretical prediction for closed-end finite disk flow.

There is a continuing debate about the role of linear instability waves in the prediction of jet noise. Parallel mean flow models, such as the one proposed by Lilley, usually neglect these waves because they cause the solution to become infinite. The result is then non-causal and can, therefore, be quite different from the more realistic causal solution, especially for the chaotic flows being considered here. The present paper solves the relevant acoustic equations for a non-parallel mean flow by using a vector Green's function approach and requiring that the mean flow be weakly non-parallel, i.e. requiring that the spread rate be small. It demonstrates that linear instability waves must be accounted for in order to construct a proper causal solution to the jet noise problem. Recent experimental results (e.g. see Tam, Golebiowski & Seiner 1996) show that the small-angle spectra radiated by supersonic jets are significantly different from those radiated at larger angles (say, at 90°) and even exhibit dissimilar frequency scalings (i.e. they scale with Helmholtz number as opposed to Strouhal number). The present solution is (among other things) able to explain this rather puzzling experimental result.

Lattice-Boltzmann simulations have been used to investigate low-Reynolds-number settling of monodisperse and polydisperse suspensions. We confirm the discovery that particle velocity fluctuations are strongly suppressed by no-slip walls at the top and bottom of the system, even in regions distant from the boundaries. We also show that a monodisperse suspension develops a strongly anisotropic long-range microstructure during the settling process, with vanishing density fluctuations in the horizontal plane. We find no numerical evidence that the particle concentration in the bulk is stratified; diffusive spreading of the suspension–supernatant interface is suppressed by hindered settling, as would be expected in moderately concentrated suspensions.

Long-range correlations in particle density fluctuations are destroyed by polydispersity in particle size, and in this case density fluctuations are finite at all length scales and in all directions. However, in polydisperse suspensions there is significant stratification, due to differential settling rather than interface diffusion, which provides an alternative mechanism for screening the hydrodynamic interactions. It is possible that this is the dominant mechanism for hydrodynamic screening in several laboratory experiments.

An analytical study of the asymptotic behaviour of descending spikes is carried out for the idealized limit of an inviscid, incompressible fluid without surface tension, bounded by a vacuum. A self-similar solution is obtained for the shape of the free surface at the spike tip, yielding the evolution in time of the surface curvature there. The approach to free-fall acceleration is shown to follow an inverse power law in time. Results are given for both planar (two-dimensional) and axisymmetric spikes. Potential areas of application include ablation-front dynamics in inertial-confinement fusion.

The ‘plug’ flow emerging from a long rotating tube into a large stationary reservoir has been used in an experimental investigation of centrifugally unstable swirling jets. A moderate Reynolds number, $\hbox{\it Re}\,{=}\,1000$, was studied extensively, and swirl numbers, $S$, the ratio of nozzle exit rotating speed to the mean mass axial velocity, were in the range 0–1.1. Four regimes were covered: non-swirling jets with $S\,{=}\,0$, weakly swirling jets in the range $0\,{<}\,S\,{<}\,S_{c1}$, strongly swirling jets before vortex breakdown with $S_{c1}\,{<}\,S\,{<}\,S_{c2}$, and stable vortex breakdown for $S\,{>}\,S_{c2}$, where $S_{c1}\,{=}\,0.6$ and $S_{c2}\,{=}\,0.88$.

Based on a general theory of detonation waves with an embedded sonic locus that we have previously developed, we carry out asymptotic analysis of weakly curved slowly varying detonation waves and show that the theory predicts the phenomenon of detonation ignition and failure. The analysis is not restricted to near Chapman–Jouguet detonation speeds and is capable of predicting quasi-steady, normal detonation shock speed versus curvature ($D$–$\kappa$) curves with multiple turning points. An evolution equation that retains the shock acceleration, $\skew2\dot{D}$, namely a $\skew2\dot{D}$–$D$–$\kappa$ relation is rationally derived which describes the dynamics of pre-existing detonation waves. The solutions of the equation for spherical detonation are shown to reproduce the ignition/failure phenomenon observed in both numerical simulations of blast wave initiation and in experiments. A single-step chemical reaction described by one progress variable is employed, but the kinetics is sufficiently general and is not restricted to Arrhenius form, although most specific calculations are performed for Arrhenius kinetics. As an example, we calculate critical energies of direct initiation for hydrogen–oxygen mixtures and find close agreement with available experimental data.

Laboratory experiments on stably stratified grid turbulence have suggested that turbulent diffusivity $\kappa_\rho$ can be expressed in terms of a turbulence activity parameter $\epsilon/\nu N^2$, with different power-law relations appropriate for different levels of $\epsilon/\nu N^2$. To further examine the applicability of these findings to both a wider range of the turbulence intensity parameter $\epsilon/\nu N^2$ and different forcing mechanisms, DNS data of homogeneous sheared stratified turbulence generated by Shih et al. (2000) and Venayagamoorthy et al. (2003) are analysed in this study. Both scalar eddy diffusivity $\kappa_\rho$ and eddy viscosity $\kappa_\nu$ are found to be well-correlated with $\epsilon/\nu N^2$, and three distinct regimes of behaviour depending on the value of $\epsilon/\nu N^2$ are apparent. In the diffusive regime $D$, corresponding to low values of $\epsilon/\nu N^2$ and decaying turbulence, the total diffusivity reverts to the molecular value; in the intermediate regime $I$, corresponding to $7 \,{<} \epsilon/\nu N^2 \,{<}\, 100$ and stationary turbulence, diffusivity exhibits a linear relationship with $\epsilon/\nu N^2$, as predicted by Osborn (1980); finally, in the energetic regime $E$, corresponding to higher values of $\epsilon/\nu N^2$ and growing turbulence, the diffusivity scales with $(\epsilon/\nu N^2)^{1/2}$. The dependence of the flux Richardson number $R_f$ on $\thing$ explains the shift in power law between regimes $I$ and $E$. Estimates for the overturning length scale and velocity scales are found for the various $\epsilon/\nu N^2$ regimes. It is noted that $\epsilon/\nu N^2 \,{\sim}\, \hbox{\it Re}/\hbox{\it Ri}\,{\sim}\,\hbox{\it ReFr}^2$, suggesting that such Reynolds–Richardson number or Reynolds–Froude number aggregates are more descriptive of stratified turbulent flow conditions than the conventional reliance on Richardson number alone.

The inviscid instability of $O(\epsilon)$ two-dimensional free-surface gravity waves propagating along an $O(1)$ parallel shear flow is considered. The modes of instability involve spanwise-periodic longitudinal vortices resembling oceanic Langmuir circulation. Here, not only are wave-induced mean effects important but also wave modulation, caused by velocity anomalies which develop in the streamwise direction. The former are described by a generalized Lagrangian-mean formulation and the latter by a modified Rayleigh equation. Since both effects are essential, the instability may be called ‘generalized’ Craik–Leibovich (CLg). Of specific interest is whether spanwise distortion of the wave field, both at the free surface and in the interior, acts to enhance or inhibit instability to longitudinal vortices. Also of interest is whether the instability gives rise to a preferred spacing for the vortices and whether that spacing concurs well or poorly with experiment. The layer depth is varied from much less than the e-folding depth of the $O(\epsilon)$ wave motion to infinity. Relative to an identical shear flow with rigid though wavy top boundary, it is found, inter alia, that wave modulation acts in concert with the free surface, at some wavenumbers, to increase the maximum growth rate of the instability. Indeed, two preferred spanwise spacings occur, one which gives rise to longitudinal vortices through a convective oscillatory bifurcation and a second, at higher wavenumber and growth rate, through a stationary bifurcation. The preferred spacings set by the stationary bifurcation concur well with those observed in laboratory experiments, with the implication that the instability acting in the experiments is very likely to be CLg.

Results of direct numerical simulation (DNS) for Rayleigh–Bénard convection for the Prandtl number $\hbox{\it Pr}\,{=}\,0.025$ are used to show some peculiarities of turbulent natural convection for low-Prandtl-number fluids. Simulations for this flow at sufficiently large Rayleigh numbers became feasible only recently because this flow requires the resolution of very small velocity scales and the recording of long-wave structures for the slow changes in the convective temperature field. The results are used to analyse standard turbulent heat flux models. The analysis for a model based on the Reynolds analogy indicates strong deficiencies of such turbulent heat flux models for low-Prandtl-number fluids. Turbulence models for buoyant flows which are not based on the Reynolds analogy include also the transport equation for the temperature variance $\overline{\theta^2}$. Detailed analysis of this transport equation and of the transport equation for the temperature variance dissipation rate is performed using DNS data. The results show the relevance of the turbulent diffusion terms and strong quantitative and qualitative deficiencies of standard models for turbulent diffusion of the temperature variance $\overline{\theta^2}$ and for the turbulent diffusion of the temperature variance dissipation rate $\epsilon_\theta$. Using the two-point correlation technique, statistical turbulence models for the turbulent diffusion of the temperature variance and for the turbulent diffusion of the temperature variance dissipation rate are proposed. These new models explicitly consider the molecular fluid properties. The new models reproduce the DNS results for $\hbox{\it Pr}\,{=}\,0.025$ and $\hbox{\it Pr}\,{=}\,0.71$ sufficiently well.

The role of coherent and incoherent subgrid-scale modes in large-eddy simulation modelling is examined. The coherent/incoherent decomposition of the subgrid-scale stresses based on the wavelet de-noising procedure is introduced. A priori dynamical tests based on the perfect modelling approach are performed for decaying isotropic turbulence. The theoretical effects of coherent and incoherent subgrid-scale forces are dynamically evaluated during the simulation. The relation between deterministic/stochastic subgrid-scale models and coherent/incoherent subgrid-scale stresses is discussed. The main result is that in large-eddy simulations low-order statistics can be almost exactly reproduced when only the effect of the coherent subgrid-scale modes is accounted for, while the incoherent subgrid-scale modes have a negligible effect upon the large-scale dynamics and the energy transfer.

Crossflow-dominated swept-wing boundary layers are known to undergo a highly nonlinear transition process. In low-disturbance environments, the primary instability of these flows consists mainly of stationary streamwise vortices that modify the mean velocity field and hence the stability characteristics of the boundary layer. The result is amplitude saturation of the dominant stationary mode and strong spanwise modulation of the unsteady modes. Breakdown is not caused by the primary instability but instead by a high-frequency secondary instability of the shear layers of the distorted mean flow. The secondary instability has been observed in several previous experiments and several computational models for its behaviour exist. None of the experiments has been sufficiently detailed to allow either model validation or transition correlation. The present experiment conducted using a 45° swept wing in the low-disturbance Arizona State University Unsteady Wind Tunnel addresses the secondary instability in a detailed fashion under a variety of conditions. The results reveal that this instability is active in the breakdown of all cases investigated, and furthermore, it appears to be well-described by the computational models.

The problem of water shipping is studied by assuming two-dimensional flow conditions and using both experimental and numerical tools. Experimentally, the water on deck for a fixed barge-shaped structure has been analysed. Video images of the water-shipping events were recorded, wave elevation in the wave flume and pressure on a vertical superstructure along the ‘ship’ deck have been measured. Numerically, a boundary element method for unsteady nonlinear free-surface flows was developed and used for the analysis of water-on-deck phenomena. A comprehensive comparison between experimental and numerical data gave satisfactory agreement globally. The synergic experimental–numerical analysis highlights the main flow features during the water shipping and details of the water impact with the deck structures are discussed. In the model tests, the water on deck started as a plunging wave hitting the deck and entrapping air. This could be relevant for deck safety, but appears to be less important for the global evolution of the water along the deck and the later liquid interaction with the superstructure. The green-water loads on the vertical wall showed a two-peak behaviour typical of wave impacts.

A linear stability analysis is conducted for the density-driven flow of variable viscosity miscible fluids in a vertically oriented capillary tube. The main goal is to assess the competition between the axisymmetric and the first azimuthal mode as a function of Rayleigh number, viscosity ratio and interfacial thickness parameter. In the absence of a net flow, the symmetry properties of the linearized set of equations indicate that the growth rates do not depend on which of the two fluids is the more viscous, although the shape of the eigenmodes does. For most parameter combinations, the first azimuthal mode is found to have larger growth rates than the axisymmetric mode. For thin interfaces and large Rayleigh numbers, however, the axisymmetric mode dominates above a certain viscosity ratio. An unexpected result is found regarding the influence of the interface thickness on the instability. For large viscosity ratios, intermediate interface thicknesses are found to be more unstable than either very thin or very thick interfaces. The reason for this behaviour is traced to a shift of the eigenfunctions towards the less viscous fluid, which allows the instability to grow in an overall less viscous environment. In the presence of a net axial flow, the upward and downward displacements of a more viscous fluid by a less viscous one are seen to result in the same growth rate. For large viscosity ratios, the axisymmetric mode becomes destabilized by the net flow, whereas the leading azimuthal mode is stabilized. This trend is in line with experimental observations.

The effect of the dispersed to continuous-phase viscosity ratio on the flow-induced coalescence of two equal-sized drops with clean interfaces was experimentally investigated. The experimental systems consisted of polybutadiene drops suspended in polydimethylsiloxane. The bulk-phase rheological properties of the fluids are Newtonian under the very weak flow conditions of the coalescence experiment (strain rate, $G \,{<}\, 0.08\,{\rm s}^{-1})$. Both head-on and glancing collisions were studied in a purely extensional flow (flow-type parameter, $\alpha\,{=}\,1.0$) for the viscosity ratio $(\lambda)$ range from $O(0.1)$ to $O(10)$. For head-on collisions, the dimensionless drainage times increased with the capillary number (Ca) as $\textit{Ca}^{3/2}$ for all the viscosity ratios, which is consistent with theoretical predictions based on a simple film drainage model. The drainage time at a fixed Ca increased with the viscosity ratio and scaled as $\lambda^{0.82}$. In the case of glancing collisions, the critical coalescence conditions were examined by changing the initial offset, which results in different collision trajectories. In an earlier paper (Yang et al. 2001) that studied a system with a viscosity ratio of 0.096, the critical capillary number $(\textit{Ca}_{c})$ for coalescence always decreased with the increasing offset. However, the present study shows that when the viscosity ratio is greater than $O(0.1)$, the critical capillary number decreases with increasing offset only for the smallest offsets, but then increases with increasing offset until a critical offset is reached above which coalescence is not observed. This is because coalescence for the larger offsets occurs in the extensional quadrant $(\phi\,{>}\,45^\circ$) after the external flow has begun to pull the drops apart. At small offsets, drops coalesced in the compression quadrant with an orientation angle, $\phi \,{<}\,45^\circ$. At the larger offsets, drops also coalesced in the compression quadrant for small Ca, but above some critical Ca, the coalescence angle jumped abruptly (i.e. with a very small change in Ca) to coalescence in the extensional quadrant. Coalescence with $\phi\,{>}\,45^\circ$ is more prevalent for the higher viscosity ratio systems. On the other hand, the maximum offset for coalescence decreased with the viscosity ratio as expected.