Results are presented from an experimental study of shallow flow in a channel partially obstructed by an array of circular cylinders. The cylinder array is a model for emergent vegetation in an open channel, but also represents a simple sparse porous medium. A shear layer with regular vortex structures forms at the edge of the array, evolving downstream to an equilibrium width and vortex size. The vortices induce nearly periodic oscillations with a frequency that matches the most unstable linear mode for a parallel shear flow. The shear layer is asymmetric about the array interface and has a two-layer structure. An inner region of maximum shear near the interface contains a velocity inflection point and establishes the penetration of momentum into the array. An outer region, resembling a boundary layer, forms in the main channel, and establishes the scale of the vortices. The vortex structure, educed by conditional sampling, shows strong crossflows with sweeps from the main channel and ejections from the array, which create significant momentum and mass fluxes across the interface. The sweeps maintain the coherent structures by enhancing shear and energy production at the interface. A linear stability analysis is consistent with the experimental results and demonstrates that the instability is excited by the differential drag between the channel and the array.

The dynamic interaction of a shockwave (modelled as a pressure pulse) with an initially spherically oscillating bubble is investigated. Upon the shockwave impact, the bubble deforms non-spherically and the flow field surrounding the bubble is determined with potential flow theory using the boundary-element method (BEM). The primary advantage of this method is its computational efficiency. The simulation process is repeated until the two opposite sides of the bubble surface collide with each other (i.e. the formation of a jet along the shockwave propagation direction). The collapse time of the bubble, its shape and the velocity of the jet are calculated. Moreover, the impact pressure is estimated based on water-hammer pressure theory. The Kelvin impulse, kinetic energy and bubble displacement (all at the moment of jet impact) are also determined. Overall, the simulated results compare favourably with experimental observations of lithotripter shockwave interaction with single bubbles (using laser-induced bubbles at various oscillation stages). The simulations confirm the experimental observation that the most intense collapse, with the highest jet velocity and impact pressure, occurs for bubbles with intermediate size during the contraction phase when the collapse time of the bubble is approximately equal to the compressive pulse duration of the shock wave. Under this condition, the maximum amount of energy of the incident shockwave is transferred to the collapsing bubble. Further, the effect of the bubble contents (ideal gas with different initial pressures) and the initial conditions of the bubble (initially oscillating vs. non-oscillating) on the dynamics of the shockwave-bubble interaction are discussed.

A database of transitional direct numerical simulation (DNS) realizations of a supercritical mixing layer is analysed for understanding small-scale behaviour and examining subgrid-scale (SGS) models duplicating that behaviour. Initially, the mixing layer contains a single chemical species in each of the two streams, and a perturbation promotes roll-up and a double pairing of the four spanwise vortices initially present. The database encompasses three combinations of chemical species, several perturbation wavelengths and amplitudes, and several initial Reynolds numbers specifically chosen for the sole purpose of achieving transition. The DNS equations are the Navier-Stokes, total energy and species equations coupled to a real-gas equation of state; the fluxes of species and heat include the Soret and Dufour effects. The large-eddy simulation (LES) equations are derived from the DNS ones through filtering. Compared to the DNS equations, two types of additional terms are identified in the LES equations: SGS fluxes and other terms for which either assumptions or models are necessary. The magnitude of all terms in the LES conservation equations is analysed on the DNS database, with special attention to terms that could possibly be neglected. It is shown that in contrast to atmospheric-pressure gaseous flows, there are two new terms that must be modelled: one in each of the momentum and the energy equations. These new terms can be thought to result from the filtering of the nonlinear equation of state, and are associated with regions of high density-gradient magnitude both found in DNS and observed experimentally in fully turbulent high-pressure flows. A model is derived for the momentum-equation additional term that performs well at small filter size but deteriorates as the filter size increases, highlighting the necessity of ensuring appropriate grid resolution in LES. Modelling approaches for the energy-equation additional term are proposed, all of which may be too computationally intensive in LES. Several SGS flux models are tested on an a priori basis. The Smagorinsky (SM) model has a poor correlation with the data, while the gradient (GR) and scale-similarity (SS) models have high correlations. Calibrated model coefficients for the GR and SS models yield good agreement with the SGS fluxes, although statistically, the coefficients are not valid over all realizations. The GR model is also tested for the variances entering the calculation of the new terms in the momentum and energy equations; high correlations are obtained, although the calibrated coefficients are not statistically significant over the entire database at fixed filter size. As a manifestation of the small-scale supercritical mixing peculiarities, both scalar-dissipation visualizations and the scalar-dissipation probability density functions (PDF) are examined. The PDF is shown to exhibit minor peaks, with particular significance for those at larger scalar dissipation values than the mean, thus significantly departing from the Gaussian behaviour.

This study provides a detailed exploration of the near-field shear-layer instabilities associated with a gaseous jet injected normally into crossflow, also known as the transverse jet. Jet injection from nozzles which are flush as well as elevated with respect to the tunnel wall are explored experimentally in this study, for jet-to-crossflow velocity ratios R in the range 1 ≲ R ≤ 10 and with jet Reynolds numbers of 2000 and 3000. The results indicate that the nature of the transverse jet instability is significantly different from that of the free jet, and that the instability changes in character as the crossflow velocity is increased. Dominant instability modes are observed to be strengthened, to move closer to the jet orifice, and to increase in frequency as crossflow velocity increases for the regime 3.5 < R ≤ 10. The instabilities also exhibit mode shifting downstream along the jet shear layer for either nozzle configuration at these moderately high values of R. When R is reduced below 3.5 in the flush injection experiments, single-mode instabilities are dramatically strengthened, forming almost immediately within the shear layer in addition to harmonic and subharmonic modes, without any evidence of mode shifting. Under these conditions, the dominant and initial mode frequencies tend to decrease with increasing crossflow. In contrast, the instabilities in the elevated jet experiments are weakened as R is reduced below about 4, probably owing to an increase in the vertical coflow magnitude exterior to the elevated nozzle, until R falls below 1.25, at which point the elevated jet instabilities become remarkably similar to those for the flush injected jet. Low-level jet forcing has no appreciable influence on the shear-layer response when these strong modes are present, in contrast to the significant influence of low-level forcing otherwise. These studies suggest profound differences in transverse-jet shear-layer instabilities, depending on the flow regime, and help to explain differences previously observed in transverse jets controlled by strong forcing.

This work concerns the breakup of millimetre-scale liquid droplets in gaseous flow fields that are disturbed from free-stream conditions by the presence of solid obstacles or other drops. A broad range of flow conditions is considered – from subsonic to supersonic, from highly rarefied to ambient pressures, and from fixed cylindrical obstacles to free liquid droplets (as obstacles). The liquid is water or tributyl phosphate, a water-like low-viscosity fluid of very low vapour pressure. We present data on deformation and breakup regimes, and, aided by numerical simulations, we discuss governing mechanisms and the time scaling of these events. Thereby a methodology is demonstrated for conveniently forecasting first-order behaviours in disturbed flow fields more generally. The highly resolved images lend themselves to testing/benchmarking numerical simulations of interfacial flows. These results, along with the experimental capability developed, constitute one of the key building blocks for our overall long-term aim towards predicting ultimate particle-size distributions from such intense aerodynamic interactions involving very large quantities of Newtonian and viscoelastic liquids.

Direct numerical simulation is used to investigate effects of heat release and compressibility on mixing-layer turbulence during a period of self-similarity. Temporally evolving mixing layers are analysed at convective Mach numbers between 0.15 and 1.1 and in a Reynolds number range of 15000 to 35000 based on vorticity thickness. The turbulence inhibiting effects of heat release are traced back to mean density variations using an analysis of the fluctuating pressure field based on a Green's function.

We consider the evolution of an annular two-dimensional region occupied by viscous fluid driven by surface tension and applied pressure at the free surfaces. We assume that the thickness of the domain is small compared with its circumference, so that it may be described as a thin viscous sheet whose ends are joined to form a closed loop. Analytical and numerical solutions of the resulting model are obtained and we show that it is well posed whether run forwards or backwards in time. This enables us to determine, in many cases explicitly, which initial shapes will evolve into a desired final shape. We also show how the application of an internal pressure may be used to control the evolution.

This work is motivated by the production of non-axisymmetric capillary tubing via the Vello process. Molten glass is fed through a die and drawn off vertically, while the shape of the cross-section evolves under surface tension and any applied pressure as it flows downstream. Here the goal is to determine the die shape required to achieve a given desired final shape, typically square or rectangular. We conclude by discussing the role of our two-dimensional model in describing the three-dimensional tube-drawing process.

We study some effects of regular bottom corrugations on water waves in a long rectangular tank with vertical endwalls and open top. In particular, we consider motions which are normal modes of oscillation in such a tank. Attention is focused on the modes whose internodal spacing, in the absence of corrugations, would be near the wavelength of the corrugations. In these cases, the perturbation of the eigenfunctions (though not of their frequencies) can be significant, e.g. the amplitude of the eigenfunction can be greater by a factor of ten or more near one end of the tank than at the other end. This is due to a cooperative effect of the corrugations, called Bragg resonance. We first study these effects using an asymptotic theory, which assumes that the bottom corrugations are of small amplitude and that the motions are slowly varying everywhere. We then present an exact theory, utilizing continued fractions. This allows us to deal with the rapidly varying components of the flow. The exact theory confirms the essential correctness of the asymptotic results for the slowly varying aspects of the motions. The rapidly varying parts (evanescent waves) are, however, needed to satisfy accurately the true boundary conditions, hence of importance to the flow near the endwalls.

This paper demonstrates that there is a contour dynamics formulation for the evolution of uniform vortex patches in any finitely connected planar domain bounded by impenetrable walls. A general numerical scheme is presented based on this formulation. The algorithm makes use of conformal mappings and follows the evolution of a conformal pre-image of a given vortex patch in a canonical multiply connected circular pre-image region. The evolution of vortex patches can be computed given just the conformal map from this pre-image region to the physical fluid region. The efficacy of the scheme is demonstrated by illustrative examples.

This paper considers the evolution of smooth, two-dimensional vortices subject to a rotating external strain field, which generates regions of recirculating, cat's eye stream line topology within a vortex. When the external strain field is smoothly switched off, the cat's eyes may persist, or they may disappear as the vortex relaxes back to axisymmetry. A numerical study obtains criteria for the persistence of cat's eyes as a function of the strength and time scale of the imposed strain field, for a Gaussian vortex profile.

In the limit of a weak external strain field and high Reynolds number, the disturbance decays exponentially, with a rate that is linked to a Landau pole of the linear inviscid problem. For stronger strain fields, but not strong enough to give persistent cat's eyes, the exponential decay of the disturbance varies: as time increases the decay slows down, because of the nonlinear feedback on the mean profile of the vortex. This is confirmed by determining the decay rate given by the Landau pole for these modified profiles. For strain fields strong enough to generate persistent cat's eyes, their location and rotation rate are determined for a range of angular velocities of the external strain field, and are again linked to Landau poles of the mean profiles, modified through nonlinear effects.

We study the growth of small-scale inhomogeneities in the density of particles floating in weakly nonlinear small-amplitude surface waves. Despite the small amplitude, the accumulated effect of the long-time evolution may produce a strongly inhomogeneous distribution of the floaters: density fluctuations grow exponentially with a small but finite exponent. We show that the exponent is of sixth or higher order in wave amplitude. As a result, the inhomogeneities do not form within typical time scales of the natural environment. We conclude that the turbulence of surface waves is weakly compressible and alone it cannot be a realistic mechanism of the clustering of matter on liquid surfaces.

It is generally accepted that ocean internal solitary waves can arise from the interaction of the barotropic tide with the continental shelf, which generates an internal tide that in turn steepens and forms solitary waves as it propagates shorewards. Some field observations, however, reveal large-amplitude internal solitary waves in deep water, hundreds of kilometres away from the continental shelf, suggesting an alternative generation mechanism: tidal flow over steep topography forces a propagating beam of internal tidal wave energy which impacts the thermocline at a considerable distance from the forcing site and gives rise to internal solitary waves there. Motivated by this possibility, a simple nonlinear long-wave model is proposed for the interaction of a tidal wave beam with the thermocline and the ensuing local generation of solitary waves. The thermocline is modelled as a density jump across the interface of a shallow homogeneous fluid layer on top of a deep uniformly stratified fluid, and a finite-amplitude propagating internal wave beam of tidal frequency in the lower fluid is assumed to be incident and reflected at the interface. The induced weakly nonlinear long-wave disturbance on the interface is governed in the far field by an integral-differential equation which accounts for nonlinear and dispersive effects as well as energy loss owing to radiation into the lower fluid. Depending on the strength of the thermocline and the intensity of the incident beam, nonlinear wave steepening can overcome radiation damping so a series of solitary waves may arise in the thermocline. Sample numerical solutions of the governing evolution equation suggest that this mechanism is quite robust for typical oceanic conditions.

The purpose of this study is to analyse the transport and stirring of fluid that occurs owing to the formation and growth of a laminar vortex ring. Experimental data was collected upstream and downstream of the exit plane of a piston-cylinder apparatus by particle-image velocimetry. This data was used to compute Lagrangian coherent structures to demonstrate how fluid is advected during the transient process of vortex ring formation. Similar computations were performed from computational fluid dynamics (CFD) data, which showed qualitative agreement with the experimental results, although the CFD data provides better resolution in the boundary layer of the cylinder. A parametric study is performed to demonstrate how varying the piston-stroke length-to-diameter ratio affects fluid entrainment during formation. Additionally, we study how regions of fluid are stirred together during vortex formation to help establish a quantitative understanding of the role of vortical flows in mixing. We show that identification of the flow geometry during vortex formation can aid in the determination of efficient stirring. We compare this framework with a traditional stirring metric and show that the framework presented in this paper is better suited for understanding stirring/mixing in transient flow problems. A movie is available with the online version of the paper.

This article deals with the first Hopf bifurcation of a cylinder flow, and more particularly with the properties of the unsteady periodic Kármán vortex street regime that sets in for supercritical Reynolds numbers Re > 46. Barkley (Europhys. Lett. vol.75, 2006, p. 750) has recently studied the linear properties of the associated mean flow, i.e. the flow which is obtained by a time average of this unsteady periodic flow. He observed, thanks to a global mode analysis, that the mean flow is marginally stable and that the eigenfrequencies associated with the global modes of the mean flow fit the Strouhal to Reynolds experimental function well in the range 46 < Re < 180. The aim of this article is to give a theoretical proof of this result near the bifurcation. For this, we do a global weakly nonlinear analysis valid in the vicinity of the critical Reynolds number Rec based on the small parameter ε = Rec−1 − Re−1 ≪ 1. We compute numerically the complex constants λ and μ′ which appear in the Stuart-Landau amplitude equation: dA/dt = ε λA − εμ′ A|A|2. Here A is the scalar complex amplitude of the critical global mode. By analysing carefully the nonlinear interactions yielding the term μ′, we show for the cylinder flow that the mean flow is approximately marginally stable and that the linear dynamics of the mean flow yields the frequency of the saturated Stuart-Landau limit cycle. We will finally show that these results are not general, by studying the case of the bifurcation of an open cavity flow. In particular, we show that the mean flow in this case remains strongly unstable and that the frequencies associated with the eigenmodes do not exactly match those of the nonlinear unsteady periodic cavity flow. It will be demonstrated that two precise conditions must hold for a linear stability analysis of a mean flow to be relevant and useful.

The origin of the stationary frontal wave packet spontaneously generated in rotating and stably stratified vortex dipoles is investigated through high-resolution three-dimensional numerical simulations of non-hydrostatic volume-preserving flow under the f-plane and Boussinesq approximations. The wave packet is rendered better at mid-depths using ageostrophic quantities like the vertical velocity or the vertical shear of the ageostrophic vertical vorticity. The analysis of the origin of vertical velocity anomalies in shallow layers using the generalized omega-equation reveals that these anomalies are related to the material rate of change of the ageostrophic differential vorticity, which in shallow layers are themselves related to the large-scale ageostrophic flow along the dipole axis, and in particular, to the advective acceleration. It is found that on the anticyclonic side of the dipole axis the combined effect of the speed and centripetal accelerations causes an anticyclonic rotation of the horizontal ageostrophic vorticity vector in a time scale of about one inertial period. These facts support the hypothesis that the origin of the stationary and spontaneously generated frontal wave packet at mid-depths is the large acceleration of the fluid particles as they move along the anticyclonic side of the dipole axis in shallow layers.

We simulate the rise of Newtonian drops in a nematic liquid crystal parallel to the far-field molecular orientation. The moving interface is computed in a diffuse-interface framework, and the anisotropic rheology of the liquid crystal is represented by the Leslie–Ericksen theory, regularized to permit topological defects. Results reveal interesting coupling between the flow field and the orientational field surrounding the drop, especially the defect configuration. The flow generally sweeps the point and ring defects downstream, and may transform a ring defect into a point defect. The stability of these defects and their transformation are depicted in a phase diagram in terms of the Ericksen number and the ratio between surface anchoring and bulk elastic energies. The nematic orientation affects the flow field in return. Drops with planar anchoring on the surface rise faster than those with homeotropic anchoring, and the former features a vortex ring in the wake. These are attributed to the viscous anisotropy of the nematic. With homeotropic anchoring, the drop rising velocity experiences an overshoot, owing to the transformation of the initial surface ring defect to a satellite point defect. With both types of anchoring, the drag coefficient of the drop decreases with increasing Ericksen number as the flow-alignment of the nematic orientation reduces the effective viscosity of the liquid crystal.

The wind-driven stably stratified mid-latitude oceanic surface turbulent boundary layer is computationally simulated in the presence of a specified surface gravity-wave field. The gravity waves have broad wavenumber and frequency spectra typical of measured conditions in near-equilibrium with the mean wind speed. The simulation model is based on (i) an asymptotic theory for the conservative dynamical effects of waves on the wave-averaged boundary-layer currents and (ii) a boundary-layer forcing by a stochastic representation of the impulses and energy fluxes in a field of breaking waves. The wave influences are shown to be profound on both the mean current profile and turbulent statistics compared to a simulation without these wave influences and forced by an equivalent mean surface stress. As expected from previous studies with partial combinations of these wave influences, Langmuir circulations due to the wave-averaged vortex force make vertical eddy fluxes of momentum and material concentration much more efficient and non-local (i.e. with negative eddy viscosity near the surface), and they combine with the breakers to increase the turbulent energy and dissipation rate. They also combine in an unexpected positive feedback in which breaker-generated vorticity seeds the creation of a new Langmuir circulation and instigates a deep strong intermittent downwelling jet that penetrates through the boundary layer and increases the material entrainment rate at the base of the layer. These wave effects on the boundary layer are greater for smaller wave ages and higher mean wind speeds.

We analyse the time response of a gas confined in a small-scale gap (of the order of or smaller than the mean free path) to an instantaneous jump in the temperature of its boundaries. The problem is formulated for a collisionless gas in the case where the relative temperature jump at each wall is small and independent of the other. An analytic solution for the probability density function is obtained and the respective hydrodynamic fields are calculated. It is found that the characteristic time scale for arriving at the new equilibrium state is of the order of several acoustic time scales (the ratio of the gap width to the most probable molecular speed of gas molecules). The results are compared with direct Monte Carlo simulations of the Boltzmann equation and good agreement is found for non-dimensional times (scaled by the acoustic time) not exceeding the system Knudsen number. Thus, the present analysis describes the early-time behaviour of systems of arbitrary size and may be useful for prescribing the initial system behaviour in counterpart continuum-limit analyses.

Long ducts (or pipes) composed of transpiring (e.g. porous) walls are at the root of numerous industrial devices for species separation, as tangential filtration or membrane desalination. Similar configurations can also be involved in fluid supply systems, as irrigation or biological fluids in capillaries. A transverse leakage (or permeate flux), the strength of which is assumed to depend linearly on local pressure (as in Starling's law for capillary), takes place through permeable walls. All other dependences, as osmotic pressure or partial fouling due to polarization of species concentration, are neglected. To analyse this open problem we consider the simplest situation: the steady laminar flow in a two-dimensional channel composed of two symmetrical porous walls.

First, dimensional analysis helps us to determine the relevant parameters. We then revisit the Berman problem that considers a uniform crossflow (i.e. pressure-independent leakage). We expand the solution in a series of Rt, the transverse Reynolds number. We note this series has a rapid convergence in the considered range of Rt (i.e. Rt ≤ O(1)). A particular method of variable separation then allows us to derive from the Navier–Stokes equations two new ordinary differential equations (ODE), which correspond to first and second orders in the development in Rt, whereas the zero order recovers the Regirer linear theory. Finally, both new ODEs are used to study the occurrence of two undesirable events in the filtration process: axial flow exhaustion (AFE) and crossflow reversal (CFR). This study is compared with a numerical approach.

The unexpected behaviour of long-runout landslides has been a controversial subject of discussion in the geophysics community. In order to provide new insight into this phenomenon, we investigate the apparent reduction of friction resulting from the presence of a second species of smaller particles in the bulk of the granular material that forms the avalanche. Results obtained by means of a two-dimensional soft particle discrete element numerical simulation are presented. The numerical experiments consider an avalanche of two-size particles, originally placed over an inclined plane. The friction coefficient for the particle–particle and wall–particle contacts is held fixed. The granular mass is allowed to evolve with time, until it comes back to rest on a horizontal plane. The position of the centre of mass is located, such that the runout length Lcm/Hcm could be measured, with Lcm and Hcm being the horizontal distance travelled and the height lost by the avalanche centre of mass, respectively. Many simulations were performed keeping the area of the avalanche constant, varying only the area fraction of small particles. The results show that the runout length increases with the area fraction of small particles, reaching a maximum for a given area fraction of small particles. A detailed analysis of the particle distribution in the granular mass indicates that the apparent friction coefficient is affected by the formation of a layer of small particles at the base of the avalanche. This layer is identified as the source of ‘lubrication’. Furthermore, since there is a dependence of the runout on the fall height and the volume in real avalanches, some simulations with different areas and different fall heights were performed. The results show a tendency of the runout to increase with area, and to decrease with the initial fall height, which is in agreement with what is observed for geological events.