We derive and investigate numerically a reduced model for wave–vortex interactions involving non-dispersive waves, which we study in a two-dimensional shallow water system with an eye towards applications in atmosphere–ocean fluid dynamics. The model consists of a coupled set of nonlinear partial differential equations for the Lagrangian-mean velocity and the wave-related pseudomomentum vector field defined in generalized Lagrangian-mean theory. It allows for two-way interactions between the waves and the balanced flow that is controlled by the distribution of Lagrangian-mean potential vorticity, and for strong solutions it features a desirable exact energy conservation law for the sum of wave energy and mean flow energy. Our model goes beyond standard ray tracing as we can derive weak solutions that contain discontinuities in the pseudomomentum field, using the theory of weakly hyperbolic systems. This allows caustics to form without predicting infinite wave amplitudes, as would be the case in the standard ray-tracing theory. Suitable wave forcing and dissipation terms are added to the model and a numerical scheme for the model is implemented as a coupled set of pseudo-spectral and finite-volume integrators. Idealized examples of interactions between wavepackets and simple vortex structures are presented to illustrate the model dynamics. The unforced and non-dissipative simulations suggest a heuristic rule of ‘greedy’ waves, i.e. in the long run the wave field always extracts energy from the mean flow.

We present the results of a set of experiments designed to measure the dispersion of non-spherical particles in a wave–current flow. We released negatively buoyant discs, rods and unit-aspect-ratio cylinders into a flow both with and without waves and analysed their respective landing positions to quantify how much they had dispersed while in the flow. We found that the presence of waves significantly increased the dispersion of the particles, and that the magnitude of this increase depends on particle shape and volume. In particular, thinner rods and thinner discs have greater relative dispersion than thicker rods and thicker discs, respectively, and smaller particles have greater relative dispersion than larger particles. Although the particles travelled farther in the presence of waves, the increase in dispersion was much greater than could be explained solely by increased transport distance. These results indicate that models of microplastic transport must account for waves as well as particle characteristics.

In a stably stratified shear layer, multiple competing instabilities produce sensitivity to small changes in initial conditions, popularly called the butterfly effect (as a flapping wing may alter the weather). Three ensembles of 15 simulated mixing events, identical but for small perturbations to the initial state, are used to explore differences in the route to turbulence, the maximum turbulence level and the total amount and efficiency of mixing accomplished by each event. Comparisons show that a small change in the initial state alters the strength and timing of the primary Kelvin–Helmholtz instability, the subharmonic pairing instability and the various three-dimensional secondary instabilities that lead to turbulence. The effect is greatest in, but not limited to, the parameter regime where pairing and the three-dimensional secondary instabilities are in strong competition. Pairing may be accelerated or prevented; maximum turbulence kinetic energy may vary by up to a factor of 4.6, flux Richardson number by 12 %–15 % and net mixing by a factor of 2.

Forcing dense suspensions of non-cohesive particles through constrictions might result in a continuous flow, an intermittent one, or indefinite interruption of flow, i.e. a clog. While one of the most important (and obvious) controlling parameters in such a system is the neck-to-particle size ratio, the role of the liquid driving method is not so straightforward. On the one hand, widespread volume-controlled systems such as syringe pumps result in pressure and local liquid velocity increases upon eventual clogs. On the other hand, pressure-controlled systems result in a decrease of the flow through the constriction when a clog is formed. The root of the question therefore lies in the role of interparticle liquid flow and hydrodynamic forces on both the formation and stability of an arch blocking the particle transport through a constriction. In this work, we study experimentally a suspension of non-cohesive particles flowing through a constricted channel (with neck-to-particle size ratio $3.03\leq D/d\leq 5.26$) in an intermittent fashion, in which they are most sensitive to parametric changes. Due to the stochastic nature of the intermittency, we make use of statistical distributions of arrest times and of discharged particles, and surprisingly, we find that the transport of non-cohesive suspensions through constrictions actually follows a ‘slower is faster’ principle under pressure-controlled driving: low imposed pressures yield intermittent non-persistent clogs, while high imposed pressures result in longer-lasting clogs, eventually becoming everlasting, and thus reducing the net particle transport rate.

This paper studies the instability of two-dimensional magnetohydrodynamic systems on a sphere using analytical methods. The underlying flow consists of a zonal differential rotation and a toroidal magnetic field is present. Semicircle rules that prescribe the possible domain of the wave velocity in the complex plane for general flow and field profiles are derived. The paper then sets out an analytical study of the ‘clamshell instability’, which features field lines on the two hemispheres tilting in opposite directions (Cally, Sol. Phys., vol. 199, 2001, pp. 231–249). An asymptotic solution for the instability problem is derived for the limit of weak shear of the zonal flow, via the method of matched asymptotic expansions. It is shown that when the zonal flow is solid body rotation, there exists a neutral mode that tilts the magnetic field lines, referred to as the ‘tilting mode’. A weak shear of the zonal flow excites the critical layer of the tilting mode, which reverses the tilting direction to form the clamshell pattern and induces the instability. The asymptotic solution provides insights into properties of the instability for a range of flow and field profiles. A remarkable feature is that the magnetic field affects the instability only through its local behaviour in the critical layer.

Turbine wake and farm blockage effects may significantly impact the power produced by large wind farms. In this study, we perform large-eddy simulations (LES) of 50 infinitely large offshore wind farms with different turbine layouts and wind directions. The LES results are combined with the two-scale momentum theory (Nishino & Dunstan, J. Fluid Mech., vol. 894, 2020, p. A2) to investigate the aerodynamic performance of large but finite-sized farms as well. The power of infinitely large farms is found to be a strong function of the array density, whereas the power of large finite-sized farms depends on both the array density and turbine layout. An analytical model derived from the two-scale momentum theory predicts the impact of array density very well for all 50 farms investigated and can therefore be used as an upper limit to farm performance. We also propose a new method to quantify turbine-scale losses (due to turbine–wake interactions) and farm-scale losses (due to the reduction of farm-average wind speed). They both depend on the strength of atmospheric response to the farm, and our results suggest that, for large offshore wind farms, the farm-scale losses are typically more than twice as large as the turbine-scale losses. This is found to be due to a two-scale interaction between turbine wake and farm induction effects, explaining why the impact of turbine layout on farm power varies with the strength of atmospheric response.

The vortex force map method for incompressible viscous flows with multiple bodies is derived in this work. The method breaks the fluid force into inertial, vortex-pressure, viscous-pressure and skin-friction components, and it could be used to analyse the fluid dynamic forces on individual bodies in a multi-body assembly. For the first time, we provide a graphical representation of the vortex-pressure force – the vortex force map – for individual bodies in a multi-body assembly. We have shown that the vortex force map in a multi-body set-up differs from single-bodied counterparts from modifications to their hypothetical potential through a nonlinear cross-coupling, and that the inertial and viscous-pressure contributions contain influences from other bodies explicitly. We then demonstrate the multi-body vortex force decomposition method with a wing–flap starting flow problem using computational fluid dynamics data, identifying the positive and negative force-generating critical regions or directions. It is found that the dominant force is the vortex-pressure force, and the force variation against time is closely related to the evolution of the vortex structures. Furthermore, we showed that the existence of another body significantly alters the force contribution roles of vortices in the flow.

In this study, we demonstrate an efficient approach to investigating the interaction of vortex pairs on a turbulent boundary layer. Our aim is to assess how vortex characteristics impact the downstream flow. Wall-modelled large eddy simulations are used together with an inlet defined by a Batchelor vortex model superimposed on a turbulent boundary layer profile, generated using synthetic turbulence and a precursor Reynolds-averaged Navier–Stokes calculation. This set-up allows for the efficient testing of multiple configurations whilst providing adequate resolution of outer boundary layer and the vortex–vortex interactions. After validating the methodology, we report a set of simulations for both co- and counter-rotating vortex pairs at different separations and asymmetric strengths. The separation distance between the vortices was found to have a significant effect upon the merging of the vortices. Asymmetric strength vortex pairs, analogous to vortex generators in yaw, demonstrate performance independent of flow direction for small angles. Detailed analysis of this flow provides insight into turbulence generation mechanisms and Reynolds stress anisotropy – valuable reference data for the development of lower-order models. Skin-friction enhancement is shown to be more effective for counter-rotating vortex pairs than co-rotating pairs of the same strength and spacing. Additionally, a wider spacing between the initial vortex positions results in a faster rise in the skin-friction coefficient.

We develop a model and numerical method to study the large-amplitude flutter of rectangular membranes (of zero bending rigidity) that shed a trailing vortex-sheet wake in a three-dimensional (3-D) inviscid fluid flow. We apply small initial perturbations and track their decay or growth to large-amplitude steady-state motions. For 12 combinations of boundary conditions at the membrane edges we compute the stability thresholds and the subsequent large-amplitude dynamics across the three-parameter space of membrane mass density, pretension and stretching rigidity. With free side edges we find good agreement with previous 2-D results that used different discretization methods. We find that the 3-D dynamics in the 12 cases naturally forms four groups based on the conditions at the leading and trailing edges. The deflection amplitudes and oscillation frequencies have scalings similar to those in the 2-D case. The conditions at the side edges, although generally less important, may have small or large qualitative effects on the membrane dynamics – e.g. steady vs unsteady, periodic vs chaotic or the variety of spanwise curvature distributions – depending on the group and the physical parameter values.

This article presents our exploration into how a finite-width internal gravity wave beam is modified by triadic resonance instability. We present both experimental and weakly nonlinear modelling to examine this instability mechanism, in which a primary wave beam generates two secondary wave beams of lower frequencies and shorter length scales. Through a versatile experimental set-up, we examine how this instability evolves over hundreds of buoyancy periods. Unlike predictions from previous zero-dimensional weakly nonlinear theory, we find that the wave does not monotonically approach a saturated equilibrium of triadic interactions; rather, the amplitudes and structures of the constituent beams continue to modulate without ever reaching a steady equilibrium. To understand this behaviour, we develop a weakly nonlinear approach to account for the spatiotemporal evolution of the amplitudes and structures of the beams over slow time scales and long distances, and explore the consequences using a numerical scheme to solve the resulting equations. Through this approach, we establish that the evolution of the instability is remarkably sensitive to the spatiotemporal triadic configuration for the system and how part of the observed modulations can be attributed to a competition between the linear growth rate of the secondary wave beams and the finite residence time of the triadic perturbations within the underlying primary beam.

A ubiquitous arrangement in nature is a free-flowing fluid coupled to a porous medium, for example a river or lake lying above a porous bed. Depending on the environmental conditions, thermal convection can occur and may be confined to the clear fluid region, forming shallow convection cells, or it can penetrate into the porous medium, forming deep cells. Here, we combine three complementary approaches – linear stability analysis, fully nonlinear numerical simulations and a coarse-grained model – to determine the circumstances that lead to each configuration. The coarse-grained model yields an explicit formula for the transition between deep and shallow convection in the physically relevant limit of small Darcy number. Near the onset of convection, all three of the approaches agree, validating the predictive capability of the explicit formula. The numerical simulations extend these results into the strongly nonlinear regime, revealing novel hybrid configurations in which the flow exhibits a dynamic shift from shallow to deep convection. This hybrid shallow-to-deep convection begins with small, random initial data, progresses through a metastable shallow state and arrives at the preferred steady state of deep convection. We construct a phase diagram that incorporates information from all three approaches and depicts the regions in parameter space that give rise to each convective state.

Motivated by drag-based propulsion of crinoids, the shape reconfiguration of a feather-like elastic structure under both steady and unsteady translational motions is investigated. The simplified elastic structure consists of a centre rod to which numerous side flaps are attached by elastic hinges. These side flaps fold in only one direction to realize a dramatic reduction in the area of the structure during the recovery stroke. Compared with experimental measurements, analytical methods developed to couple the dynamics of the centre rod and the side flaps successfully predict the drag force and three-dimensional reconfiguration of the elastic structure during both power and recovery strokes. A dimensionless speed given by the ratio of inertial fluid force to elastic bending force is proposed for the coupled deflections of the centre rod and side flaps, and is found to determine primarily the reconfiguration of the elastic structure. A reconfiguration number defined specifically for our model provides an appropriate characterization of the effect of side-flap folding on drag force reduction. Moreover, the ratio of drag forces between the power and recovery strokes is evaluated to find model conditions for the optimal force ratio.

The presence of a no-slip, impermeable, adiabatic, sloped boundary in an otherwise quiescent, stably stratified, Boussinesq flow generates baroclinic vorticity within a diffusive boundary layer. Such conditions are typical of the oscillating boundary layers on adiabatic abyssal slopes, sloped lake bathymetry and sloped coastal bathymetry in the absence of high-wavenumber internal waves, mean flows, far-field turbulence on larger scales, and resonant tidal–bathymetric interaction. We investigate the linear stability of the oscillating flow within non-dimensional parameter space typical of the $M_2$ tide and hydraulically smooth, middle-latitude abyssal slopes through Floquet linear stability analysis. The flow dynamics depends on three non-dimensional variables: the Reynolds number for Stokes’ second problem, the Prandtl number, and a frequency ratio that accounts for the resonance conditions ($C$, criticality) of the buoyant restoring force and the tidal forcing. The Floquet analysis results suggest that oscillating laminar boundary layers on adiabatic abyssal slopes are increasingly unstable as Reynolds number, criticality parameter and/or spanwise disturbance wavenumber are increased. We also show that the two-dimensional Floquet linear instability necessarily generates three-dimensional baroclinic vorticity, which suggests that the evolution of the gravitational instabilities may be nonlinear as $t\rightarrow \infty$.

A small spheroid settling in a quiescent fluid experiences an inertial torque that aligns it so that it settles with its broad side first. Here we show that an active particle experiences such a torque too, as it settles in a fluid at rest. For a spherical squirmer, the torque is $\boldsymbol {T}^\prime = -{\frac {9}{8}} m_f (\boldsymbol {v}_s^{(0)} \wedge \boldsymbol {v}_g^{(0)})$ where $\boldsymbol {v}_s^{(0)}$ is the swimming velocity, $\boldsymbol {v}_g^{(0)}$ is the settling velocity in the Stokes approximation and $m_f$ is the equivalent fluid mass. This torque aligns the swimming direction against gravity: swimming up is stable, swimming down is unstable.

Well-resolved direct numerical simulations of turbulent open channel flows (OCFs) are performed for friction Reynolds numbers up to $Re_\tau =2000$. Various turbulent statistics are documented and compared with the closed channel flows (CCFs). As expected, the mean velocity profiles of the OCFs match well with the CCFs in the near-wall region but diverge notably in the outer region. Interestingly, a logarithmic layer with Kárman constant $\kappa =0.363$ occurs for OCF at $Re_\tau =2000$, distinctly different from CCF. Except for a very thin layer near the free surface, most of the velocity and vorticity variances match between OCFs and CCFs. The one-dimensional energy spectra reveal that the very-large-scale motions (VLSMs) with streamwise wavelength $\lambda _x>3 h$ or spanwise wavelength $\lambda _z>0.5 h$ contribute the most to turbulence intensity and Reynolds shear stress in the overlap and outer layers (where h is the water depth). Furthermore, the VLSMs in OCFs are stronger than those in CCFs, resulting in a slightly higher streamwise velocity variance in the former. Due to the footprint effect, these structures also have significant contributions to the mean wall shear stress, and the difference between OCF and CCF enlarges with increasing $Re_\tau$. In summary, the free surface in OCFs plays an essential role in various flow phenomena, including the formation of stronger VLSMs and turbulent kinetic energy redistribution.

We report a molecularly augmented continuum-based computational model of dynamic wetting and apply it to the displacement of an externally driven liquid plug between two partially wetted parallel plates. The results closely follow those obtained in a recent molecular dynamics (MD) study of the same problem (Fernández-Toledano et al., J. Colloid Interface Sci., vol. 587, 2021, pp. 311–323), which we use as a benchmark. We are able to interpret the maximum speed of dewetting $U^*_{{crit}}$ as a fold bifurcation in the steady phase diagram and show that its dependence on the true contact angle $\theta _{{cl}}$ is quantitatively similar to that found using MD. A key feature of the model is that the contact angle is dependent on the speed of the contact line, with $\theta _{{cl}}$ emerging as part of the solution. The model enables us to study the formation of a thin film at dewetting speeds $U^*>U^*_{{crit}}$ across a range of length scales, including those that are computationally prohibitive to MD simulations. We show that the thickness of the film scales linearly with the channel width and is only weakly dependent on the capillary number. This work provides a link between matched asymptotic techniques (valid for larger geometries) and MD simulations (valid for smaller geometries). In addition, we find that the apparent angle, the experimentally visible contact angle at the fold bifurcation, is not zero. This is in contrast to the prediction of conventional treatments based on the lubrication model of flow near the contact line, but consistent with experiment.

Tandem cylinder flow comprises several different flow regimes. Within the reattachment regime, the development of the gap shear layers is of utmost importance to the flow, but has received little attention so far. Through direct numerical simulations at $Re = 10^{4}$, for a gap ratio of 3.0, we have discovered that the shear layers are significantly altered with respect to a single cylinder. These differences include early onset of separation, crossflow stabilising, delayed transition to turbulence and little meandering of the transition region. Vortex pairing in the gap shear layers is reported for the first time. The interaction between the recirculating gap flow and the shear layers was investigated. Asymmetrical, large-scale gap vortices influence the position of transition to turbulence through direct contact and through secondary flows. The occurrence of transition in the gap shear layers has consequences for both the reattachment mechanism and the development of the downstream cylinder wake. The reattachment points are unsteady with large amplitude fluctuations on a fine time scale. Reattachment is seen to be a combination of impingement and modification of the upstream shear layers, which causes a double shear layer in the downstream cylinder near-wake. Buffeting by and interaction with the gap shear layers likely cause transition to turbulence in the downstream cylinder boundary layer. This leads to significant changes in the wake topology, compared with a single-cylinder wake.

The motion of a long gas bubble in a confined capillary tube is ubiquitous in a wide range of engineering and biological applications. While the understanding of the deposited thin viscous film near the tube wall in Newtonian fluids is well developed, the deposition dynamics in commonly encountered non-Newtonian fluids remains much less studied. Here, we investigate the dynamics of a confined bubble moving in shear-thinning fluids with systematic experiments, varying the zero-shear-rate capillary number $Ca_0$ in the range of $O(10^{-3}\unicode{x2013}10^2)$ considering the zero-shear-rate viscosity. The thickness of the deposited liquid film, the bubble speed and the bubble front/rear menisci are measured, which are further rationalized with the recent theoretical studies based on appropriate rheological models. Compared with Newtonian fluids, the film thickness decreases for both the carboxymethyl cellulose and Carbopol solutions when the shear-thinning effect dominates. We show that the film thickness follows the scaling law from Aussillous & Quéré (Phys. Fluids, vol. 12, no. 10, 2000, pp. 2367–2371) with an effective capillary number $Ca_e$, considering the characteristic shear rate in the film as proposed by Picchi et al. (J. Fluid Mech., vol. 918, no. A7, 2021, pp. 1–30). $Ca_e$ is calculated by the Carreau number and the power-law index from the Carreau–Yasuda rheological model. The shear-thinning effect also influences the bubble speed and delays the transition to the parabolic region in the bubble front and rear menisci. In particular, a high degree of undulations on the bubble surface results in an intricate rear viscosity distribution for the rear meniscus and the deviation between the experiments and theory may require a further investigation to resolve the axial velocity field. Our study may advance the fundamental understandings and engineering guidelines for coating processes involving thin-film flows and non-Newtonian fluids.

An intriguing experimental observation in electrocoalescence of water-in-oil emulsions is the occurrence of a very low critical electric field, beyond which chaining of droplets and shorting of electrodes is observed, as compared with the experimental and theoretical predictions based on two equal sized water droplets in oil. Motivated by these observations, a numerical, analytical and experimental study on the interaction between multiple, unequal sized, perfectly conducting droplets in a perfectly dielectric medium under an electric field is presented here. We show that the critical capillary number ($Ca_c$), based on the bigger droplet, in a two droplet system, reduces as the radius ratio of the smaller to bigger drop decreases. Secondly, in a system of three equally sized droplets, it is expected that the $Ca_c$ will be smaller than a two equal sized droplet system, since the electric field experienced by the central droplet is higher when surrounded by two droplets instead of one. Our results show that nonlinearity in the system due to both the asymmetric shape deformation and the electrostatic interaction between the multiple droplets, leads to significant reduction in $Ca_c$ for onset of non-coalescence in an unequal sized two droplet system or for equal and unequal sized three droplet systems, as compared with $Ca_c$ for two equal sized droplets. This is possibly one of the underlying mechanisms for observing much smaller $Ca_c$ in emulsions as compared with a system of two equal sized droplets, and could be responsible for a polydisperse water-in-oil emulsion being exceptionally susceptible to chaining under an electric field.

When making directional surface gravity waves in a wave tank or when initialising numerical simulations of the ocean, the wave spectrum is often curtailed suppressing higher frequencies and wavenumbers. We consider the impact of doing this by numerically simulating two seminal experiments, those of Onorato et al. (J. Fluid Mech., vol. 627, 2009, pp. 235–257, R2) and Latheef & Swan (Proc. R. Soc. A, vol. 469, no. 2152, 2013, p. 20120696). We simulate waves using a fully nonlinear potential-flow model. We find that curtailing the spectrum can have a significant impact on the subsequent evolution. In particular, for cases where the spectrum has been curtailed, the nonlinear physics produces significantly more extreme or rogue waves than are observed in the case where the full spectral tail was included in the initial conditions, and this difference persists over tens of periods after the waves are initialised. This suggests that sea states that are ‘out of equilibrium’ (i.e. with their tails removed) can produce a greater number of rogue waves. We show this can also have an impact on predicted loads on offshore infrastructure.