Similarly to their biological counterparts, suspensions of chemically active autophoretic swimmers exhibit a non-trivial dynamics involving self-organisation processes as a result of inter-particle interactions. Using a kinetic model for a dilute suspension of autochemotactic Janus particles, we analyse the effect of a confined pressure-driven flow on these collective behaviours and the impact of chemotactic aggregation on the effective viscosity of the active fluid. Four dynamic regimes are identified when increasing the strength of the imposed pressure-driven flow, each associated with a different collective behaviour resulting from the competition of flow- and chemically induced reorientation of the swimmers together with the constraints of confinement. Interestingly, we observe that the effect of the pusher (respectively puller) hydrodynamic signature, which is known to reduce (respectively increase) the effective viscosity of a sheared suspension, is inverted upon the emergence of autochemotactic aggregation. Our results provide new insights into the role of the collective dynamics in complex environments, which are relevant to synthetic as well as biological systems.

The Green's function approach of Giles and Pierce (J. Fluid Mech., vol. 426, 2001, pp. 327–345) is used to build the lift and drag based analytic adjoint solutions for the two-dimensional incompressible Euler equations around irrotational base flows. The drag-based adjoint solution turns out to have a very simple closed form in terms of the flow variables and is smooth throughout the flow domain, while the lift-based solution is singular at rear stagnation points and sharp trailing edges owing to the Kutta condition. This singularity is propagated to the whole dividing streamline (which includes the incoming stagnation streamline and the wall) upstream of the rear singularity (trailing edge or rear stagnation point) by the sensitivity of the Kutta condition to changes in the stagnation pressure.

The effects of viscoelasticity have been shown to manifest themselves via symmetry breaking. In this investigation, we show a novel phenomenon that arises from this idea. We observe that when a dense sphere is rotated near a wall (the rotation being aligned with the wall-normal direction and gravity), it levitates to a fixed distance away from the wall. Since the shear is larger in the gap (between the sphere and the wall) than in the open side of the sphere, the shear-induced elastic stresses are thus asymmetric, resulting in a net elastic vertical force that balances the weight of the sphere. We conduct experiments, theoretical models and numerical simulations for rotating spheres of various sizes and densities in a Boger-type fluid. In the small-Deborah-number range, the results are collapsed into a universal trend by considering a dimensionless group of the ratio of elastic to gravitational forces.

In this paper, the instability of shallow-water shear flow with a sheared parallel magnetic field is studied. Waves propagating in such magnetic shear flows encounter critical levels where the phase velocity relative to the basic flow, $c-U(y)$, matches the Alfvén wave velocities $\pm B(y)/\sqrt {\mu \rho }$, based on the local magnetic field $B(y)$, the magnetic permeability $\mu$, and the mass density of the fluid $\rho$. It is shown that when the two critical levels are close to each other, the critical layer can generate an instability. The instability problem is solved, combining asymptotic solutions at large wavenumbers and numerical solutions, and the mechanism of instability explained using the conservation of momentum. For the shallow-water magnetohydrodynamic system, the paper gives the general form of the local differential equation governing such coalescing critical layers for any generic field and flow profiles, and determines precisely how the magnetic field modifies the purely hydrodynamic stability criterion based on the potential vorticity gradient in the critical layer. The curvature of the magnetic field profile, or equivalently the electric current gradient $J' = - B''/\mu$ in the critical layer, is found to play a complementary role in the instability.

The atomization of a water column by a gas jet flow (Reynolds number $\sim O(10^{4}\unicode{x2013}10^{5})$) issued from a two-stage annular nozzle is investigated experimentally. Varying the nozzle geometry, the momentum flux ratio of the upper and lower jets, and the water flow rate, we measure the processes of atomization with high-speed imaging, analysed analytically into four regimes. In the bulk atomization regime, the atomization is driven by the lower jet, but it is forced to occur earlier by the stronger upper jet before the water column reaches the lower jet in the droplet atomization regime. Interestingly, the size of the atomized droplets remains unaffected by the momentum flux ratio of upper to lower jets. The atomization process is governed by the Rayleigh–Taylor instability, by which the estimated droplet size agrees well with the measurement. In the backflow regime, a strong reverse flow is induced to force a substantial portion of atomized droplets to be drawn backward to the nozzle; a floating liquid column regime is captured transitionally, i.e. the column stagnates near the lower nozzle when the water flow rate is very low. To understand the mechanisms of each regime, the single-phase jet flow is measured separately using particle image velocimetry, and implemented into the control volume analysis with which we predicted analytically and validated the conditions for the occurrence of each regime. It is found that the acceleration of gas flow (velocity gradient) experienced by the falling water is the key parameter to drive the atomization.

We investigate the statistics of rogue waves occurring in the inverse cascade of surface gravity wave turbulence. In such statistically homogeneous, stationary and isotropic wave fields, low-frequency waves are generated by nonlinear interactions rather than directly forced by a wave maker. This provides a laboratory realization of arguably the simplest nonlinear sea state, in which long-time acquisitions are performed and compared with theoretical models. The analysis of thousands of rogue waves reveals that some of their properties crucially depend on four-wave resonant interactions, large crests being for instance more likely than predicted by second-order models.

We study experimentally the behaviour of negatively buoyant disks and fibres in a turbulent boundary layer. The regime is relevant to the transport of natural sediment or plastic particles in water, with density ratio $\rho _p/\rho _f \sim O(1)$, major axis lengths $D_p^+ \sim 50$, friction Stokes numbers $\mbox { {St}}^+ \sim O(10)$ and friction Reynolds number $\mbox { {Re}}_\tau = 620$. The translational and rotational motion, as well as concentration and dispersion, are compared with those of spheres of similar inertia. Disks and fibres both oversample high-speed fluid near the wall, in agreement with particle-resolved numerical simulations. Fibres tend to orient mostly in the streamwise direction while disks maintain their symmetry axis quasi-normal to the wall. This alignment is more stable for disks than for fibres: the latter undergo strong tumbling near the wall in response to the mean shear and turbulent fluid velocity fluctuations, whereas the former wobble about their preferential wall-normal orientation. The translational and rotational accelerations indicate that, despite the nominal relaxation times being similar, the disks are slower than the fibres in responding to wall turbulence. For both, wall contact causes strong and intermittent tumbling. The concentration profiles follow Rouse–Prandtl theory over a limited portion of the boundary layer, deviating near the wall and in the outer region. This is largely due to the non-uniform settling velocity, which decreases steeply approaching the wall for all particle types. This is, in turn, a consequence of the reduced particle diffusivity, which closely matches the profile of the eddy viscosity.

Container motion along a planar circular trajectory at a constant angular velocity, i.e. orbital shaking, is of interest in several industrial applications, e.g. for fermentation processes or in cultivation of stem cells, where good mixing and efficient gas exchange are the main targets. Under these external forcing conditions, the free surface typically exhibits a primary steady-state motion through a single-crest dynamics, whose wave amplitude, as a function of the external forcing parameters, shows a Duffing-like behaviour. However, previous experiments in laboratory-scale cylindrical containers have revealed that, owing to the excitation of super-harmonics, diverse dynamics are observable in certain driving-frequency ranges. Among these super-harmonics, the double-crest dynamics is particularly relevant, as it displays a notably large amplitude response, which is strongly favoured by the spatial structure of the external forcing. In the inviscid limit and with regards to circular cylindrical containers, we formalize here a weakly nonlinear analysis via a multiple-time-scale method of the full hydrodynamic sloshing system, leading to an amplitude equation suitable for describing such a double-crest swirling motion. The weakly nonlinear prediction is shown to be in fairly good agreement with previous experiments described in the literature. Lastly, we discuss how an analogous amplitude equation can be derived by solving asymptotically for the first super-harmonic of the forced Helmholtz–Duffing equation with small nonlinearities.

We present a versatile perturbative calculation scheme to determine the leading-order correction to the mobility matrix for particles in a low-Reynolds-number fluid with spatially variant viscosity. To this end, we exploit the Lorentz reciprocal theorem and the reflection method in the far field approximation. To demonstrate how to apply the framework to a particular choice of a viscosity field, we first study particles in a finite-size, interface-like, linear viscosity gradient. The extent of the latter should be significantly larger than the particle separation. Both situations of symmetrically and asymmetrically placed particles within such an odd symmetric viscosity gradient are considered. As a result, long-range flow fields are identified that decay by one order slower than their constant-viscosity counterparts. Self-mobilities for particle rotations and translations are affected, while for asymmetric placement, additional correction appears for the latter. The mobility terms associated with hydrodynamic interactions between the particles also need to be corrected, in a placement-specific manner. While the results are derived for the system of two particles, they apply also to many-particle systems. Furthermore, we treat the viscosity gradients induced by two particles with temperatures different from that of the surrounding fluid. Assuming a linear relation between fluid temperature and viscosity, we find that both the self-mobilities of the particles as well as the mobility terms for hydrodynamic interactions increase for hot particles and decrease for cold particles.

Flow-induced vibrations (FIV) of an elastically mounted circular cylinder are investigated by means of two-dimensional simulations. A mechanical coupling between cross-flow translation and rotation provides a single degree-of-freedom system in which the coupled rotational oscillations affect the fluid–structure dynamics. The structural response of this system is investigated exploring the design space spanned by reduced velocity, coupling radius and phase density ratio. The kinematic coupling introduces the rotation-induced shear layer modifications, as well as an equivalent inertia effect connected to the coupling force. Such a computational campaign is carried out by means of direct numerical simulations with immersed boundary forcing at a Reynolds number equal to 100. The investigated system exhibits the wake-body synchronisation features typical of lock-in for non-rotating cylinders. However, the kinematic coupling provides a novel FIV scenario, in which the oscillation amplitude is magnified in the locked configurations with respect to the forced rotation case. Furthermore, it is found that there a significant widening of the reduced velocity domain where the lock-in condition takes place. In view of the proposed analyses, it is determined that the coupled rotation guarantees the phase alignment between lift and displacement necessary to sustain the lock-in condition, making the oscillation amplitude grow indefinitely with the reduced velocity. This is inherently achieved due to the rotational shear layer and the added mass contribution, which prevent the exact match between oscillation frequency and system natural frequency in vacuum. The outcomes of this study might potentially lead to an innovative water energy harvester offering larger power outputs and extended optimal operating regions.

Ocean gravity currents flow along the inclined ocean floor for long times compared to the planet's rotation period. Their shape and motion is governed by the gravity buoyancy, Coriolis acceleration and friction-induced Ekman-layer spinup circulation. In order to understand this process, we consider the flow of a dense-fluid Boussinesq gravity current of fixed volume over an inclined bottom in a rotating system, in the framework of Cartesian 2.5-dimensional geometry (no dependency on the lateral direction $y$, but with a non-trivial $y$-component velocity $v$ due to Coriolis coupling with the main $u$ along the bottom $x$). After release from rest in a lock (co-rotating, with two gates creating propagation in ${\pm }x$-directions), the current forms a quasi-steady geostrophic ‘vein’ of parabolic height profile with a significant lateral velocity $v$. Subsequently, a spinup process, driven by the Ekman layers on the bottom and interface, appears and prevails for many revolutions, during which $v$ decays and the shape of the interface changes dramatically. We investigate the spinup motion, using an approximate model, for the case of large Rossby number, small Ekman number and small slope $\gamma$ (relevant to oceanic currents). We show that the initial shape of the natural geostrophic vein can be calculated rigorously (not an arbitrary parabola), and the initial lateral velocity $v(x,t=0)$ is counter-rotation about a fixed point (pivot) $x_{\rm \pi}$ at which $v(x_{\rm \pi},t) =0$ (at the beginning and during spinup). This point is placed excentrically, in the upper part, and this excentre, $\propto \gamma$, plays a significant role in the process. The spinup in a rigid container is developed as the prototype process; an essential component is the edge (outer wall) where the flux of the Ekman layer is arrested (and then returned to the centre via the inviscid core). While the upper part of the vein adopts this spinup pattern, the lower part (most of the vein, $x< x_{\rm \pi}$) develops a leak (drainage) at the edge that (a) modifies the spinup of the vein, and (b) generates a thin tail extension downslope. The tail consists of two merged non-divergent Ekman layers, which chokes the drainage flow rate. The present model provides clear-cut insights and some quantitative predictions of the major spinup stage by analytical algebraic solutions. A comparison with a previously published simple model (Wirth, Ocean Dyn., vol. 59, 2009, pp. 551–563) is presented. We also discuss briefly stability of the initial vein.

Partially wetting droplets under an airflow can exhibit complex behaviours that arise from the coupling of surface tension, inertia of the external flow and contact-line dynamics. Recent experiments by Hooshanginejad et al. (J. Fluid Mech., vol. 901, 2020) revealed that a millimetric partially wetting water droplet under an impinging jet can oscillate in place, split or depin away from the jet, depending on the magnitude (i.e. $5\unicode{x2013}20\ {\rm m}\ {\rm s}^{-1}$) and position of the jet. To rationalise the experimental observations, we develop a two-dimensional lubrication model of the droplet that incorporates the external pressure of the impinging high-Reynolds-number jet, in addition to the capillary and hydrostatic pressures of the droplet. Distinct from the previous model by Hooshanginejad et al. (J. Fluid Mech., vol. 901, 2020), we simulate the motion of the contact line using precursor film and disjoining pressure, which allows us to capture a wider range of droplet behaviours, including the droplet dislodging to one side. Our simulations exhibit a comparable time-scale of droplet deformations and similar outcomes as the experimental observations. We also obtain the analytical steady-state solutions of the droplet shapes and construct the minimum criteria for splitting and depinning.

Weakly nonlinear internal wave–wave interaction is a key mechanism that cascades energy from large to small scales, leading to ocean turbulence and mixing. Oceans typically have a non-uniform density stratification profile; moreover, submarine topography leads to a spatially varying bathymetry ($h$). Under these conditions and assuming mild-slope bathymetry, we employ multiple-scale analysis to derive the wave amplitude equations for weakly nonlinear wave–wave interactions. The waves are assumed to have a slowly (rapidly) varying amplitude (phase) in space and time. For uniform stratifications, the horizontal wavenumber ($k$) condition for waves (1, 2, 3), given by ${k}_{(1,a)}+{k}_{(2,b)}\!+\!{k}_{(3,c)}\!=\!0$, is unaffected as $h$ is varied, where $(a,b,c)$ denote the mode number. Moreover, the nonlinear coupling coefficients (NLC) are proportional to $1/h^2$, implying that triadic waves grow faster while travelling up a seamount. For non-uniform stratifications, triads that do not satisfy the condition $a=b=c$ may not satisfy the horizontal wavenumber condition as $h$ is varied, and unlike uniform stratification, the NLC may not decrease (increase) monotonically with increasing (decreasing) $h$. NLC, and hence wave growth rates for weakly nonlinear wave–wave interactions, can also vary rapidly with $h$. The most unstable daughter wave combination of a triad with a mode-1 parent wave can also change for relatively small changes in $h$. We also investigate higher-order self-interactions in the presence of a monochromatic, small-amplitude topography; here, the topography behaves as a zero-frequency wave. We derive the amplitude evolution equations and show that higher-order self-interactions might be a viable mechanism of energy cascade.

A two-dimensional model for the evolution of the fire line – the interface between burned and unburned regions of a wildfire – is formulated. The fire line normal velocity has three contributions: (i) a constant rate of spread representing convection and radiation effects; (ii) a curvature term that smooths the fire line; and (iii) a Stefan-like term in the direction of the oxygen gradient. While the first two effects are geometrical, (iii) is dynamical and requires the solution of the steady advection–diffusion equation for oxygen, with advection owing to a self-induced ‘fire wind’, modelled by the gradient of a harmonic potential field. The conformal invariance of this coupled pair of partial differential equations, which has the Péclet number $\textit {Pe}$ as its only parameter, is exploited to compute numerically the evolution of both radial and infinitely long periodic fire lines. A linear stability analysis shows that fire line instability is possible, dependent on the ratio of curvature to oxygen effects. Unstable fire lines develop finger-like protrusions into the unburned region; the geometry of these fingers is varied and depends on the relative magnitudes of (i)–(iii). It is argued that for radial fires, the fire wind strength scales with the fire's effective radius, meaning that $\textit {Pe}$ increases in time, so all fire lines eventually become unstable. For periodic fire lines, $\textit {Pe}$ remains constant, so fire line stability is possible. The results of this study provide a possible explanation for the formation of fire fingers observed in wildfires.

This paper presents bifurcation analyses characterising the nonlinear dynamics of fully developed laminar annular jets with respect to the centrebody diameter ${ {d}}$, Reynolds number ${ {Re}}$, and swirl ratio ${ {S}}$. Similar flows appear in numerous applications and feature a vibrant range of topological and dynamical characteristics associated with phenomena including shear layer separation and vortex breakdown. Our results begin by describing the non-monotonic evolution of the axisymmetric jet's steady topology under varying ${ {S}}$. In accord with earlier reports, the jet progresses through a sequence of wake, breakdown and wall jet regimes in a qualitatively similar manner across a wide span of ${ {d}}$ and ${ {Re}}$ values. In the wake regime, the non-swirling jet bifurcates to a plane-symmetric, but not axisymmetric, steady flow pattern beyond a ${ {d}}$-dependent critical ${ {Re}}$ value. With further increase in ${ {Re}}$, the steady non-swirling jet destabilises subsequently via multiple distinct Hopf bifurcations. Introducing ${ {S}}>0$ to the jet also induces unsteadiness by twisting the singly azimuthally periodic ($|m|=1$) asymmetric wake structure and causing it to precess periodically in time about the central axis. Intermediate swirl stabilises this unsteady dynamics and restores the jet's axisymmetry. This stabilising effect is then reversed in the breakdown regime at higher ${ {S}}$, where a variety of different $|m|=1$ and $|m|=2$ instabilities bifurcate from the steady flow as ${ {S}}$ is increased. Several instances of hysteresis and subcritical behaviour are reported and discussed, including one that manifests precessing vortex core oscillations.

The rules that govern a two-dimensional inviscid incompressible fluid are simple. Yet, to characterise the long-time behaviour is a knotty problem. The fluid fulfils Euler's equations: a nonlinear Hamiltonian system with an infinite number of conservation laws. In both experiments and numerical simulations, coherent vortex structures emerge after an initial stage. These formations dominate the large-scale dynamics, but small scales also emerge and persist. The resulting scale separation resembles Kraichnan's theory of forward and backward cascades of enstrophy and energy. Previous attempts to model the double cascade use filtering techniques that enforce separation from the outset. Here, we show that Euler's equations possess an intrinsic, canonical splitting of the vorticity function. The splitting is remarkable in four ways: (i) it is defined solely by the Poisson bracket and the Hamiltonian; (ii) it characterises steady flows; (iii) it innately separates scales, enabling the dynamics behind Kraichnan's qualitative description; and (iv) it accounts for ‘broken line’ energy spectra observed in both experiments and numerical simulations. The splitting originates from Zeitlin's truncated model of Euler's equations in combination with a standard quantum tool: the spectral decomposition of Hermitian matrices. In addition to theoretical insight, the scale separation dynamics enables stochastic model reduction, where multiplicative noise models small scales.

The changing three-dimensional vortex shedding topology of a heavy, tethered sphere exposed to uniform flow crossing the onset of vortex induced vibrations (VIV) is reported together with simultaneously tracked sphere positions. Transient upstream flow conditions were imposed by increasing the reduced velocity from $U^{*} = 2.2$ to 4.5 (‘lock-in’, mode I). Three-dimensional vortex shedding under these transient conditions strongly resembled those at steady conditions at the same $U^{*}$. Changes in the wake accompanying the onset of VIV indicated several well-defined stages associated with the changing instantaneous location of the velocity deficit centroid and the wake's symmetry plane's orientation. Before the onset of VIV, the appearance of induced vortices led to lock-in of streamwise sphere oscillations prior to lock-in in the transverse direction. Shortly after ($U^{*} \approx 3.6$), preferential vortex shedding (wake symmetry plane aligned with tether) was lost. Upon reaching $U^{*} = 4.5$, flow–structure interaction reorganised vortex shedding and sphere motion, resulting in steady state conditions after some delay. At this stage, the symmetry plane was aligned perpendicular to the tether and pairs of alternately shed, single hairpins exerted transverse forcing on the sphere. At $U^{*} = 7.2$ (mode II, steady upstream flow), pairs of double hairpins were shed per oscillation period with maximum instantaneous vortex force coefficients that were higher than at $U^{*} = 4.5$. While the present Reynolds numbers ($Re$) were chosen low, the different identified stages in the onset of VIV are also expected to be relevant at a higher $Re$ range.

The performance benefits of deploying tidal turbines in close side-by-side proximity to exploit constructive interference effects are demonstrated experimentally using two 1.2 m diameter turbines. The turbines are arrayed side-by-side at 1/4 diameter tip-to-tip spacing, and their performance compared with that of a single rotor. Tests were completed in the 25 m diameter, 2 m deep wave and current FloWave Ocean Energy Research facility. A detailed assessment of inflow conditions at different control points is used to understand the impact that rotors, designed for high blockage conditions, have on the approach flow. After accounting for global blockage, a 10.8 % uplift in the twin-turbine-averaged power coefficient, relative to that for a single turbine, is found for the turbine design speed, at the expense of a 5.2 % increase in thrust coefficient and 3.1 % increase in tip-speed-ratio. Flowfield mapping demonstrated flow effects at array and device scale including array bypass flows and jetting between turbines. Azimuthal variation of blade root flapwise and edgewise bending moments show that the turbines interact in a beneficial manner, with additional and sustained loading peaks as the blades pass in close proximity to the neighbouring rotor. Peak performance for the twin turbines occurred at a higher tip-speed-ratio than for the single turbine, which is consistent with the twin turbines exerting a higher thrust on the flow to achieve maximum power. The twin turbine performance variation with tip-speed-ratio is found to be more gradual than for the single turbine. Using differential rotor speed control we observe that array performance is robust to small differences in neighbouring rotor operating point. Through these experiments we demonstrate that there is a substantial, achievable performance benefit from closely arraying turbines for side-by-side operation and designing them for constructive interference.

The present study deals with close-contact melting of a vertical cylinder on a horizontal isothermal superhydrophobic surface with an array of circular posts. A new numerical model for this phenomenon is formulated under several simplifying assumptions. For the limiting case of perfect slip and thermal contact, based on the model assumptions, it is demonstrated that superhydrophobic surfaces can enhance the melting time by no more than 30 %. Numerical solution of the new model reveals that the effect of superhydrophobic hydrodynamic slip can decrease the molten layer thickness. However, due to the dominant effect of thermal slip, the heat transfer rate can be decreased significantly, and the melting time can be roughly doubled. An approximate analytical model is developed for understanding the problem dynamics better. The analytical model predictions are compared extensively against the numerical model results. For sufficiently low surface volume fractions, the analytical model provides excellent estimations of the numerical model predictions. Dimensional analysis reveals that according to the analytical model, the ratio between the melting times for superhydrophobic and regular surfaces is governed by a single dimensionless group. Moreover, according to the analytical model, the melting time on a superhydrophobic surface is always longer than on a regular surface, which agrees qualitatively with the numerical model results. Based on this analysis, a dimensionless condition for the analytical model range of validity is formulated. Finally, a conservative criterion for the liquid–gas interface stability under a close-contact melting process is established.

Direct numerical simulations of turbulent non-rotating and rotating plane Couette flow with a periodically modulated plate velocity are conducted to study the effect of modulated forcing on turbulent shear flows. The time-averaged shear Reynolds number is fixed at $Re_S = 3 \times 10^4$, which results in a frictional Reynolds number of approximately $Re_\tau \approx 400$. The modulating frequency is varied in the range $Wo\in (20,200)$, while the modulating amplitude is kept fixed at $10\,\%$ of the shear velocity except to demonstrate that varying this parameter has minimal effect. The resulting shear at the plates is found to be independent of the forcing frequency, and equal to the non-modulated baseline. For the non-rotating simulations, two clear flow regions can be seen: a near-wall region that follows Stokes’ theoretical solution, and a bulk region that behaves similar to Stokes’ solutions but with an increased effective viscosity. For high driving frequencies, the amplitude response follows the scaling laws for modulated turbulence of von der Heydt et al. (Phys. Rev. E, vol. 67, 2003, 046308). Cyclonic rotation is not found to modify the system's behaviour in a substantial way, but anti-cyclonic rotation changes significantly the system's response to periodic forcing. We find that the persistent axial inhomogeneities introduced by mild anti-cyclonic rotation make it impossible to measure the propagation of the modulation adequately, while stronger anti-cyclonic rotation creates regions where the modulation travels instantaneously.