In a recent paper, Chu & Prosperetti (J. Fluid Mech., vol. 798, 2016, pp. 787–811) calculate the dissolution of a two-component droplet in an immiscible liquid. Here we discuss in what sense their results go much beyond the Epstein–Plesset solution of a dissolving single-component droplet and hitherto used approximations for dissolving multicomponent droplets. We also highlight the relevance of Chu & Prosperetti’s result for liquid–liquid extraction processes for chemical analysis.

In a channel flow, the velocity fluctuations are inhomogeneous and anisotropic. Yet, the small-scale properties of the flow are expected to behave in an isotropic manner in the very-large-Reynolds-number limit. We consider the statistical properties of small-scale velocity fluctuations in a turbulent channel flow at moderately high Reynolds number ($Re_{\unicode[STIX]{x1D70F}}\approx 1000$), using the Johns Hopkins University Turbulence Database. Away from the wall, in the logarithmic layer, the skewness of the normal derivative of the streamwise velocity fluctuation is approximately constant, of order 1, while the Reynolds number based on the Taylor scale is $R_{\unicode[STIX]{x1D706}}\approx 150$. This defines a small-scale anisotropy that is stronger than in turbulent homogeneous shear flows at comparable values of $R_{\unicode[STIX]{x1D706}}$. In contrast, the vorticity–strain correlations that characterize homogeneous isotropic turbulence are nearly unchanged in channel flow even though they do vary with distance from the wall with an exponent that can be inferred from the local dissipation. Our results demonstrate that the statistical properties of the fluctuating velocity gradient in turbulent channel flow are characterized, on one hand, by observables that are insensitive to the anisotropy, and behave as in homogeneous isotropic flows, and on the other hand by quantities that are much more sensitive to the anisotropy. How this seemingly contradictory situation emerges from the simultaneous action of the flux of energy to small scales and the transport of momentum away from the wall remains to be elucidated.

I present a theoretical and experimental study of floating viscous fluid films introduced into a channel of finite length, motivated by the flow of glacial ice shelves. The dynamics are characterized by a mixture of viscous extensional stresses, transverse shear stresses and a driving buoyancy force. A theory based on a width-integrated model is developed and investigated using analytical, asymptotic and numerical methods. With fluid introduced at a constant rate, the flow is found to approach a steady state with two possible asymptotic forms depending on the length of the channel. For channel lengths less than half the width, the flow is similar to a purely extensional one-dimensional flow, characterized by concave surface profiles and being insensitive to the position of the channel exit (or calving front). Greater lengths result in a more complex asymptotic structure in which the flow adjusts over a short distance towards a prevailing flow of universal dimensionless form. In complete contrast to the extensional regime, the prevailing flow is controlled by the position of the channel exit. Data from a new laboratory experiment involving particle velocimetry of a floating fluid film compares well with the predicted along-channel velocity. Motivated by glaciological application, the analysis is generalized to power-law rheologies and the results used to classify the flow regimes of a selection of ice shelves. The prediction for the frontal speed is in good agreement with geophysical data, indicating that the universal profile predicted by the theory is common in nature.

We establish a theoretical model for the radial oscillations, translational motions and deformations of two interacting encapsulated bubbles. The flow field outside the bubbles is approximated by a potential flow with a viscous correction. The in-plane stresses and bending moments of the viscoelastic membranes are balanced by the hydrodynamic tractions at the interfaces of the bubbles. Since the material points move along the membranes accompanied by their movements in the radial direction when the encapsulated bubbles undergo deformations, stress balance in both the tangential and normal directions and the no-velocity-jump condition at the bubble surface are applied. The derived expression for the viscous drag includes the quasisteady drag force and the history force, which is validated by the solution of the unsteady Stokes equation. With an appropriate choice of the interface parameters, the present model is suitable for bubbles with free-slip, viscoelastic or no-slip interfaces. The viscous correction and the potential part of our solution are validated, respectively, by comparing them with previous experimental observations. The encapsulated bubble shows more stability in resisting shape oscillation. The attractive or repulsive movements of the two bubbles subjected to a driving frequency are consistent with the prediction by Bjerknes’ theory. For gas bubbles, the drag is mainly from the quasisteady component of the flow. For encapsulated bubbles, the no-velocity-jump condition enhances viscous dissipation, and thus contributes significantly to the history force in the viscous drag, generating more damping in the translational motion.

Recent field experiments conducted in the near wake (up to 0.5 rotor diameters downwind of the rotor) of a Clipper Liberty C96 2.5 MW wind turbine using snow-based super-large-scale particle image velocimetry (SLPIV) (Hong et al., Nat. Commun., vol. 5, 2014, 4216) were successful in visualizing tip vortex cores as areas devoid of snowflakes. The so-visualized snow voids, however, suggested tip vortex cores of complex shape consisting of circular cores with distinct elongated comet-like tails. We employ large-eddy simulation (LES) to elucidate the structure and dynamics of the complex tip vortices identified experimentally. We show that the LES, with inflow conditions representing as closely as possible the state of the flow approaching the turbine when the SLPIV experiments were carried out, reproduce vortex cores in good qualitative agreement with the SLPIV results, essentially capturing all vortex core patterns observed in the field in the tip shear layer. The computed results show that the visualized vortex patterns are formed by the tip vortices and a second set of counter-rotating spiral vortices intertwined with the tip vortices. To probe the dependence of these newly uncovered coherent flow structures on turbine design, size and approach flow conditions, we carry out LES for three additional turbines: (i) the Scaled Wind Farm Technology (SWiFT) turbine developed by Sandia National Laboratories in Lubbock, TX, USA; (ii) the wind turbine developed for the European collaborative MEXICO (Model Experiments in Controlled Conditions) project; and (iii) the model turbine presented in the paper by Lignarolo et al. (J. Fluid Mech., vol. 781, 2015, pp. 467–493), and the Clipper turbine under varying inflow turbulence conditions. We show that similar counter-rotating vortex structures as those observed for the Clipper turbine are also observed for the SWiFT, MEXICO and model wind turbines. However, the strength of the counter-rotating vortices relative to that of the tip vortices from the model turbine is significantly weaker. We also show that incoming flows with low level turbulence attenuate the elongation of the tip and counter-rotating vortices. Sufficiently high turbulence levels in the incoming flow, on the other hand, tend to break up the coherence of spiral vortices in the near wake. To elucidate the physical mechanism that gives rise to such rich coherent dynamics we examine the stability of the turbine tip shear layer using the theory proposed by Leibovich & Stewartson (J. Fluid Mech., vol. 126, 1983, pp. 335–356). We show that for all simulated cases the theory consistently indicates the flow to be unstable exactly in the region where counter-rotating spirals emerge. We thus postulate that centrifugal instability of the rotating turbine tip shear layer is a possible mechanism for explaining the phenomena we have uncovered herein.

We present a Newtonian, one-dimensional, differential analysis for capillary breakup rheometry (CBR) to determine the surface tension to viscosity ratio $\unicode[STIX]{x1D6FC}$. Our local differential analysis does not require specific assumptions for the axial force to preclude its measurement. Our analysis indicates that measuring gradients in filament curvature is necessary to accurately determine $\unicode[STIX]{x1D6FC}$ when axial force is not measured. CBR experiments were performed on five silicone oils ($0.35~\text{Pa}~\text{s}<\unicode[STIX]{x1D707}<10~\text{ Pa}~\text{s}$), three sample volumes, and three strains to evaluate the operating range of the differential analysis and compare its performance to that of a standard integral method from literature. We investigate the role of filament asymmetry, caused mainly by gravity, on the performance of the differential method for the range of conditions studied. Experimental and analytical details for resolving gradients of curvature are also given.

We investigate rough-wall turbulent flows through direct numerical simulations of flow over three-dimensional transitionally rough sinusoidal surfaces. The roughness Reynolds number is fixed at $k^{+}=10$, where $k$ is the sinusoidal semi-amplitude, and the sinusoidal wavelength is varied, resulting in the roughness solidity $\unicode[STIX]{x1D6EC}$ (frontal area divided by plan area) ranging from 0.05 to 0.54. The high cost of resolving both the flow around the dense roughness elements and the bulk flow is circumvented by the use of the minimal-span channel technique, recently demonstrated by Chung et al. (J. Fluid Mech., vol. 773, 2015, pp. 418–431) to accurately determine the Hama roughness function, $\unicode[STIX]{x0394}U^{+}$. Good agreement of the second-order statistics in the near-wall roughness-affected region between minimal- and full-span rough-wall channels is observed. In the sparse regime of roughness ($\unicode[STIX]{x1D6EC}\lesssim 0.15$) the roughness function increases with increasing solidity, while in the dense regime the roughness function decreases with increasing solidity. It was found that the dense regime begins when $\unicode[STIX]{x1D6EC}\gtrsim 0.15{-}0.18$, in agreement with the literature. A model is proposed for the limit of $\unicode[STIX]{x1D6EC}\rightarrow \infty$, which is a smooth wall located at the crest of the roughness elements. This model assists with interpreting the asymptotic behaviour of the roughness, and the rough-wall data presented in this paper show that the near-wall flow is tending towards this modelled limit. The peak streamwise turbulence intensity, which is associated with the turbulent near-wall cycle, is seen to move further away from the wall with increasing solidity. In the sparse regime, increasing $\unicode[STIX]{x1D6EC}$ reduces the streamwise turbulent energy associated with the near-wall cycle, while increasing $\unicode[STIX]{x1D6EC}$ in the dense regime increases turbulent energy. An analysis of the difference of the integrated mean momentum balance between smooth- and rough-wall flows reveals that the roughness function decreases in the dense regime due to a reduction in the Reynolds shear stress. This is predominantly due to the near-wall cycle being pushed away from the roughness elements, which leads to a reduction in turbulent energy in the region previously occupied by these events.

We derive a new model for thin viscous liquid films down an inclined plane. With an asymptotic expansion in the long-wave limit, the Navier–Stokes equations and the work–energy theorem are averaged over the fluid depth. This gives three equations for the mass, momentum and energy balance which have the mathematical structure of the Euler equations of compressible fluids with relaxation source terms, diffusive and capillary terms. The three variables of the model are the fluid depth, the average velocity and a third variable called enstrophy, related to the variance of the velocity. The equations are numerically solved by classical schemes which are known to be reliable and robust. The model gives satisfactory results both for the neutral stability curves and for the depth profiles of wavy films produced by a periodical forcing or by a random noise perturbation. The numerical calculations agree fairly well with experimental measurements of Liu & Gollub (Phys. Fluids, vol. 6, 1994, pp. 1702–1712). The calculation of the wall shear stress below the waves indicates a flow reversal at the first depth minimum downstream of the main hump, in agreement with experiments of Tihon et al. (Exp. Fluids, vol. 41, 2006, pp. 79–89).

Internal solitary waves in a system of two fluids, silicone oil and water, bounded above by a free surface are studied both experimentally and theoretically. By adjusting an extra volume of silicone oil released from a reservoir, a wide range of amplitude waves are generated in a wave tank. Wave profiles as well as wave speeds are measured using multiple wave probes and are then compared with both the weakly nonlinear Korteweg–de Vries (KdV) models and the strongly nonlinear Miyata–Choi–Camassa (MCC) models. As the density difference between the two fluids in the experiment is relatively small (approximately 14 %), but non-negligible, special attention is paid to the effect of the boundary condition at the top surface. The nonlinear models valid for rigid-lid (RL) and free-surface (FS) boundary conditions are considered separately. It is found that the solitary wave of the FS model for a given amplitude is consistently narrower than that of the RL model and it propagates at a slightly lower speed. Due to strong nonlinearity in the internal-wave motion, the weakly nonlinear KdV models fail to describe the measured internal solitary wave profiles of intermediate and large wave amplitudes. The strongly nonlinear MCC-FS model agrees better with the measurements than the MCC-RL model, which indicates that the free-surface boundary condition at the top surface is crucial in describing the internal solitary waves in the experiment correctly. Leaving the top surface free in the experiment allows us to observe small and relatively short wave packets on the top surface, particularly when the amplitude of the internal solitary wave is large. Once excited, the wave packet is located above the front half of the internal solitary wave and propagates with a speed close to that of the internal solitary wave underneath. A simple resonance mechanism between short surface waves and long internal waves without and with nonlinear effects is examined to estimate the characteristic wavelength of modulated short surface waves, which is found to be in good agreement with the observed wavelength when nonlinearity is taken into account. Using ray theory, the evolution of short surface waves in the presence of a background current induced by an internal solitary wave is also investigated to examine the location of the modulated surface wave packet.

The occurrence of the elliptic instability in rings and helical vortices is analysed theoretically. The framework developed by Moore & Saffman (Proc. R. Soc. Lond. A, vol. 346, 1975, pp. 413–425), where the elliptic instability is interpreted as a resonance of two Kelvin modes with a strained induced correction, is used to obtain the general stability properties of a curved and strained Batchelor vortex. Explicit expressions for the characteristics of the three main unstable modes are obtained as a function of the axial flow parameter of the Batchelor vortex. We show that vortex curvature adds a contribution to the elliptic instability growth rate. The results are applied to a single vortex ring, an array of alternate vortex rings and a double helical vortex.

This paper reports a numerical study of two-dimensional periodically perturbed flow past a cylinder. Both harmonic and non-harmonic perturbation waveforms of the inflow velocity are considered for a maximum instantaneous Reynolds number of 180. Phase portraits of the lift force are employed to identify the dynamical state of the cylinder wake and determine the range of kinematical parameters for which primary synchronization occurs, that is the regime where vortex formation is phase-locked to the subharmonic of the perturbation frequency. The effect of different perturbation waveforms on the synchronization range and on patterns of vortex formation is examined in detail over the normalized amplitude–frequency space. It is shown that systematic shifts of the synchronization range, towards either higher or lower frequencies, can be attained by imposing different perturbation waveforms. As a consequence, in certain regions of the parameter space it is possible to obtain multiple periodic and/or quasi-periodic wake states for waveforms of exactly the same amplitude and frequency. For the range of parameters where synchronization occurs, different vortex patterns are attained in the wake involving the shedding of solitary and binary vortices, or mixtures thereof, which can be controlled by the perturbation waveform. The phenomenology of these patterns, which result from modification of the basic Kármán mode in the unperturbed wake, is discussed and explained in terms of the generation of circulation that is induced by perturbations in the relative velocity. Vortex patterns from cylinders oscillating either in line with or transverse to a free stream are recast in the framework of the relative velocity.

The free shear layer that separates from the leading edge of a round-nosed plate has been studied under conditions of low (background) and elevated (grid-generated) free stream turbulence (FST) using high-fidelity particle image velocimetry. Transition occurs after separation in each case, followed by reattachment to the flat surface of the plate downstream. A bubble of reverse flow is thereby formed. First, we find that, under elevated (7 %) FST, the time-mean bubble is almost threefold shorter due to an accelerated transition of the shear layer. Quadrant analysis of the Reynolds stresses reveals the presence of slender, highly coherent fluctuations amid the laminar part of the shear layer that are reminiscent of the boundary-layer streaks seen in bypass transition. Instability and the roll-up of vortices then follow near the crest of the shear layer. These vortices are also present under low FST and in both cases are found to make significant contributions to the production of Reynolds stress over the rear of the bubble. But their role in reattachment, whilst important, is not yet fully clear. Instantaneous flow fields from the low-FST case reveal that the bubble of reverse flow often breaks up into two or more parts, thereby complicating the overall reattachment process. We therefore suggest that the downstream end of the ‘separation isoline’ (the locus of zero absolute streamwise velocity that extends unbroken from the leading edge) be used to define the instantaneous reattachment point. A histogram of this point is found to be bimodal: the upstream peak coincides with the location of roll-up, whereas the downstream mode may suggest a ‘flapping’ motion.

This paper treats the macro-scale description of the periodically developed conjugate heat transfer regime, in which heat transfer takes place between an incompressible viscous flow and spatially periodic solid structures through a spatially periodic interfacial heat flux. The macro-scale temperature of the fluid and the solid structures are defined through a spatial averaging operator with a specific weighting function. It is shown that a double volume average is necessary in order to have a linearly changing macro-scale temperature in response to a constant macro-scale heat flux. Furthermore, with the aid of a double volume average, the thermal dispersion source, the thermal tortuosity and the interfacial heat transfer coefficient all become spatially constant in the developed regime. That way, these closure terms of the macro-scale temperature equations can be exactly determined from the periodic temperature part on a unit cell of the solid structures without taking the spatial moments of the solid into account. The theoretical derivations of this paper are illustrated for a case study describing the heat transfer between a fluid flow and an array of solid squares with a uniform volumetric heat source.

The early nonlinear transition process initiated by a small-amplitude pair of oblique waves is studied using both temporal numerical simulation and theoretical considerations. This investigation is performed under the flow conditions of the experiments by Corke et al. (AIAA J., vol. 40, 2002, pp. 1015–1018) who investigated a sharp $7^{\circ }$ cone in the NASA Mach 3.5 Quiet Tunnel. In particular, both the linear and the nonlinear mechanisms prior to transition onset are investigated in great detail as the physics of this regime predetermine the flow topology of the nonlinear transition zone. The objective of this study is (i) to advance the understanding of the underlying physical mechanisms relevant for the early nonlinear transition regime of oblique breakdown and (ii) to make the connection to oblique transition, the incompressible scenario for bypass transition investigated by Schmid & Henningson (Phys. Fluids A, vol. 4, 1992, pp. 1986–1989). The dominance of the longitudinal vortex mode in oblique breakdown is shown to be a consequence of a constantly forced transient growth instability. In particular, the primary pair of oblique waves serves as an ‘actuator’ that is permanently introducing disturbances into the longitudinal mode where these disturbances exhibit transient growth. The effect of the transient growth instability on the longitudinal mode is to raise its amplitude rather than change the growth rate of this mode.

We report on a series of laboratory experiments in which we investigate the mixing in a wake produced downstream of an obstacle in a uniform flow. The fluid is confined within a channel of finite width, and the water depth is small compared with the channel width. The mixing appears to be dominated by dispersion caused by the circulation of the eddies that are shed alternately from each side of the obstacle. However, due to bottom friction, these eddies gradually dissipate downstream. In turn, the intensity of the cross-stream mixing of the tracer decays in the downstream direction, limiting the cross-stream extent of the tracer. We present a time-averaged picture of the experiments which illustrates the deviation of the time-averaged flow in the wake relative to the uniform flow upstream. We then develop a time-averaged model for the flow, using mixing length theory to account for the cross-channel momentum transfer as an eddy viscosity $\unicode[STIX]{x1D706}_{1}ud$, where $2ud$ is the cross-channel integral of the perturbation in the along-channel speed associated with the wake. We also include a frictional stress to account for the bottom friction. The model predicts a similar pattern of variation of the along-channel velocity in both the along- and cross-channel directions to our experimental data. By matching the cross-channel data with the model, we find that the constant $\unicode[STIX]{x1D706}_{1}$ has value 0.2. We also analyse our experimental data to develop a time-dependent picture of the mixing of a stream of dye released into the wake. Using the model for the evolution of the flow, we develop a model for the time-averaged mixing, again based on mixing length theory. The model predicts a similar spatial distribution for the tracer in both the cross-stream and along-stream directions to that seen in our experimental data. By quantitative comparison of the model with the data, we find that the best fit of the empirical eddy diffusivity, $\unicode[STIX]{x1D706}_{2}ud$, with the data occurs with $\unicode[STIX]{x1D706}_{2}=0.22$. We discuss implications of our results for modelling cross-stream mixing in shallow turbulent flow.

We consider Rayleigh–Bénard convection in a layer of fluid between rough no-slip boundaries where the top and bottom boundary heights are functions of the horizontal coordinates with square-integrable gradients. We use the background method to derive an upper bound on the mean heat flux across the layer for all admissible boundary geometries. This flux, normalized by the temperature difference between the boundaries, can grow with the Rayleigh number ($Ra$) no faster than $O(Ra^{1/2})$ as $Ra\rightarrow \infty$. Our analysis yields a family of similar bounds, depending on how various estimates are tuned, but every version depends explicitly on the boundary geometry. In one version the coefficient of the $O(Ra^{1/2})$ leading term is $0.242+2.925\Vert \unicode[STIX]{x1D735}h\Vert ^{2}$, where $\Vert \unicode[STIX]{x1D735}h\Vert ^{2}$ is the mean squared magnitude of the boundary height gradients. Application to a particular geometry is illustrated for sinusoidal boundaries.

The statistics of the velocity gradient tensor in turbulent flows is of both theoretical and practical importance. The Lagrangian view provides a privileged perspective for studying the dynamics of turbulence in general, and of the velocity gradient tensor in particular. Stochastic models for the Lagrangian evolution of velocity gradients in isotropic turbulence, with closure models for the pressure Hessian and viscous Laplacian, have been shown to reproduce important features such as non-Gaussian probability distributions, skewness and vorticity strain-rate alignments. The recent fluid deformation (RFD) closure introduced the idea of mapping an isotropic Lagrangian pressure Hessian as the upstream initial condition using the fluid deformation tensor. Recent work on a Gaussian fields closure, however, has shown that even Gaussian isotropic velocity fields contain significant anisotropy for the conditional pressure Hessian tensor due to the inherent velocity–pressure couplings, and that assuming an isotropic pressure Hessian as the upstream condition may not be realistic. In this paper, Gaussian isotropic field statistics is used to generate more physical upstream conditions for the recent fluid deformation mapping. In this new framework, known isotropy relations can be satisfied by tuning the free model parameters and the original Gaussian field coefficients can be directly used without direct numerical simulation (DNS)-based re-adjustment. A detailed comparison of results from the new model, referred to as the recent deformation of Gaussian fields (RDGF) closure, with existing models and DNS shows the improvements gained, especially in various single-time statistics of the velocity gradient tensor at moderate Reynolds numbers. Application to arbitrarily high Reynolds numbers remains an open challenge for this type of model, however.

Direct numerical simulation data from turbulent pipe and channel flows at $\mathit{Re}_{\unicode[STIX]{x1D70F}}\approx 930$ are used to investigate their statistical difference by focusing on large-scale motions (LSMs). The contribution to the bulk production of turbulent kinetic energy shows marked differences in the overlap and core regions. These discrepancies arise from the dominant contributions of the LSMs ($\unicode[STIX]{x1D706}_{x}>3\unicode[STIX]{x1D6FF}$) to the Reynolds shear stress in the channel flow. The spectrum of the net Reynolds shear force reveals that the LSMs accelerate the mean flow in the overlap region. The net force spectrum is further decomposed into the spectra of velocity–vorticity correlations, $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$, which are related to the advective vorticity transport and the change-of-scale effect, respectively. The dominance of large-scale accelerating motions (LSAMs) in the overlap region of the channel flow is due to the contribution of $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ at longer wavelengths ($\unicode[STIX]{x1D706}_{x}>3\unicode[STIX]{x1D6FF}$), The LSAMs are related to the long low-speed regions, and these regions are longer and wider in the channel flow than in the pipe flow. Due to the pipe curvature, the spanwise size of the LSMs is restricted by neighbouring LSMs and the spanwise velocity fluctuations are reduced. The contribution of $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ to the acceleration is prominent in the channel flow, leading to the dominance of the LSAMs associated with the change-of-scale effect.

Steady laminar flow over a rounded-tip $2\,:\,1$ elliptic cone of 0.86 m length at zero angle of attack and yaw has been computed at Mach number $7.45$ and unit Reynolds number $Re^{\prime }=1.015\times 10^{7}~\text{m}^{-1}$. The flow conditions are selected to match the planned flight of the Hypersonic Flight Research Experimentation HIFiRE-5 test geometry at an altitude of 21.8 km. Spatial linear BiGlobal modal instability analysis of this flow has been performed at selected streamwise locations on planes normal to the cone symmetry axis, resolving the entire flow domain in a coupled manner while exploiting flow symmetries. Four amplified classes of linear eigenmodes have been unravelled. The shear layer formed near the cone minor-axis centreline gives rise to amplified symmetric and antisymmetric centreline instability modes, classified as shear-layer instabilities. At the attachment line formed along the major axis of the cone, both symmetric and antisymmetric instabilities are also discovered and identified as boundary-layer second Mack modes. In both cases of centreline and attachment-line modes, symmetric instabilities are found to be more unstable than their antisymmetric counterparts. Furthermore, spatial BiGlobal analysis is used for the first time to resolve oblique second modes and cross-flow instabilities in the boundary layer between the major- and minor-axis meridians. Contrary to predictions for the incompressible regime for swept infinite wing flow, the cross-flow instabilities are not found to be linked to the attachment-line instabilities. In fact, cross-flow modes peak along most of the surface of the cone, but vanish towards the attachment line. On the other hand, the leading oblique second modes peak near the leading edge and their associated frequencies are in the range of the attachment-line instability frequencies. Consequently, the attachment-line instabilities are observed to be related to oblique second modes at the major-axis meridian. The linear amplification of centreline and attachment-line instability modes is found to be strong enough to lead to laminar–turbulent flow transition within the length of the test object. The predictions of global linear theory are compared with those of local instability analysis, also performed here under the assumption of locally parallel flow, where use of this assumption is permissible. Fair agreement is obtained for symmetric centreline and symmetric attachment-line modes, while for all other classes of linear disturbances use of the proposed global analysis methodology is warranted for accurate linear instability predictions.

The vectoring of a pair of parallel synthetic jets can be described using five dimensionless parameters: the aspect ratio of the slots, the Strouhal number, the Reynolds number, the phase difference between the jets and the spacing between the slots. In the present study, the influence of the latter four on the vectoring behaviour of the jets is examined experimentally, using particle image velocimetry. Time-averaged velocity maps are used to give a qualitative description of the variations in vectoring for a parametric sweep of each of the four parameters independently. A diverse set of vectoring behaviour is observed in which the resulting jet can be merged or bifurcated and either vectored towards the actuator leading in phase or the actuator lagging in phase. Three performance metrics are defined to give a quantitative description of the vectoring behaviour: the included angle between bifurcated branches, the vectoring angle of the total flow and the normalized momentum flux of the flow. Using these metrics, the influence of changes in the Strouhal number, Reynolds number, phase difference and spacing are quantified. Phase-locked maps of the swirling strength are used to track vortex pairs. Vortex trajectories are used to define three Strouhal number regimes for the vectoring behaviour. In the first regime, vectoring behaviour is dominated by the pinch-off time, which is written as function of Strouhal number only. In the second regime, the pinch-off time is invariant and the vectoring behaviour slightly changes with Strouhal number. In the third regime, given by the formation criterion, no synthetic jet is formed. Vortex positions at a single phase, shortly after creation of the lagging vortex pair, are used to propose a vectoring mechanism. This vectoring mechanism explains the observed qualitative and quantitative variations for all four parameters.