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.

Direct numerical simulations of the flow past an impulsively started cylinder at high Reynolds numbers (25k–100k) reveal an intriguing portrait of unsteady separation. Vorticity generation and vortex shedding entails a cascade of separation events on the cylinder surface that are reminiscent of Kelvin–Helmholtz instabilities. Primary vortices roll up along the cylinder surface as a result of instabilities of the initially attached vortex sheets, followed by vortex eruptions, creation of secondary vorticity and formation of dipole structures that are subsequently ejected from the surface of the cylinder. We analyse the vortical structures and their relationship to the forces experienced by the cylinder. This striking cascade of vortex instabilities may serve as reference for reduced-order models of flow separation and as guide for flow control of separated flows at high Reynolds numbers.

The problem of a plane unsteady potential flow of an ideal incompressible fluid bounded by free boundary segments with a constant pressure and by solid walls moving in accordance with a known law is considered. External forces are absent, and capillary forces are neglected. An approach to constructing exact solutions for this type of problem is proposed. The corresponding solutions can be treated as nonlinear perturbations of a certain base flow. As an example of the application of this approach, nonlinear perturbations in a known problem of a fluid flow with a linear velocity field in the region bounded by a straight-line free boundary and parallel approaching or receding solid walls are considered. It is demonstrated that perturbations grow, which leads to variants of the formation of singularities on the free surface of the fluid within a finite time: formation of droplets, bubbles or cusps. A solution describing the collapse of a bubble in a fluid layer bounded by two approaching solid walls has also been found and studied. Thus, a new method of studying nonlinear stability of complicated unsteady fluid flows with combined boundary conditions is proposed and tested.

The flow past a cylinder in proximity to a plane wall is investigated numerically for small gap ratios. Three vortex dynamic processes associated with different hairpin vortex generation mechanisms are identified for the first time, and the wake-induced turbulent transition is analysed. The vortex shedding is suppressed at $G/D = 0.1$, while the spanwise vortex is generated via a Kelvin–Helmholtz instability and evolves into hairpin vortices. For $G/D= 0.3$, the upper and lower rollers alternatively shedding from the cylinder, interact with the secondary vortex. The split secondary vortex merges with the upper roller and results in a new vortex downstream, which develops into hairpin vortices. When $G/D = 0.9$, the secondary vortex interacts with the lower roller and then evolves into hairpin vortices. A tertiary vortex induced by the secondary vortex is observed, rotating in the opposite direction to the secondary vortex the wake-induced transitions share the same route. The velocity fluctuations deviate from the optimal growth theory in the pre-transitional region. In the transitional region low-frequency disturbances penetrate the sheltering edge to generate streaks where the disturbance energy declines. In the turbulent region the logarithmic layer is formed, indicating that the turbulent equilibrium is established.

Incipient wave breaking on a vertical circular pile is simulated with a Reynolds stress–$\omega$ turbulence model. Comparison of results simulated with a stabilized two-equation turbulence model, as well as no turbulence model, demonstrates that the breaking point and the peak force on a vertical cylinder due to incipient breaking should not be affected by the turbulence closure model, provided that it is stable and the simulations are converged. Notably, the present results show that the build-up to peak force induced by incipient wave breaking can be accurately predicted without any turbulence closure model. However, for the prediction of the secondary load cycle (SLC), proper turbulence modelling is required, as this process involves both turbulence production and lee-side flow separation. The Reynolds stress–$\omega$ model is demonstrated to predict the SLC more accurately than a stabilized two-equation $k$–$\omega$ turbulence model, as the flow separation points and vorticity field are better predicted. Some existing studies indicate that the generation of the SLC does not necessarily result from flow separation, but is rather due to the suction force. The present work finds that the occurrence and point of flow separation significantly affect the magnitude of the suction force, which hence affects the SLC prediction significantly. For waves breaking on a vertical pile, proper turbulence modelling is therefore essential for accurate SLC predictions. (In the above, $k$ is the turbulent kinetic energy density and $\omega$ is the specific dissipation rate.)

We revisit the construction of discrete kinetic models for single-component isothermal two-phase flows. Starting from a kinetic model for a non-ideal fluid, we show that, under conventional scaling, the Navier–Stokes equations with a non-ideal equation of state are recovered in the hydrodynamic limit. A scaling based on the smallness of velocity increments is then introduced, which recovers the full Navier–Stokes–Korteweg equations. The proposed model is realized on a standard lattice and validated on a variety of benchmarks. Through a detailed study of thermodynamic properties including co-existence densities, surface tension, Tolman length and sound speed, we show thermodynamic consistency, well-posedness and convergence of the proposed model. Furthermore, hydrodynamic consistency is demonstrated by verification of Galilean invariance of the dissipation rate of shear and normal modes and the study of visco-capillary coupling effects. Finally, the model is validated on dynamic test cases in three dimensions with complex geometries and large density ratios such as drop impact on textured surfaces and mercury drops coalescence.

Non-isothermal particles suspended in a fluid lead to complex interactions – the particles respond to changes in the fluid flow, which in turn is modified by their temperature anomaly. Here, we perform a novel proof-of-concept numerical study based on tracer particles that are thermally coupled to the fluid. We imagine that particles can adjust their internal temperature reacting to some local fluid properties, and follow simple, hard-wired active control protocols. We study the case where instabilities are induced by switching the particle temperature from hot to cold depending on whether it is ascending or descending in the flow. A macroscopic transition from stable to unstable convective flow is achieved, depending on the number of active particles and their excess negative/positive temperature. The stable state is characterized by a flow with low turbulent kinetic energy, strongly stable temperature gradient, and no large-scale features. The convective state is characterized by higher turbulent kinetic energy, self-sustaining large-scale convection, and weakly stable temperature gradients. Individually, the particles promote the formation of stable temperature gradients, while their aggregated effect induces large-scale convection. When the Lagrangian temperature scale is small, a weakly convective laminar system forms. The Lagrangian approach is also compared to a uniform Eulerian bulk heating with the same mean injection profile, and no such transition is observed. Our empirical approach shows that thermal convection can be controlled by pure Lagrangian forcing, and opens the way for other data-driven particle-based protocols to enhance or deplete large-scale motion in thermal flows.

The impact of drops on a porous surface with high contact-angle hysteresis and gas discharge is studied. Four different impact modes, ranging from complete repulsion to fast immobilization of a drop on the surface, are identified and mapped in a space spanned by the pressure difference of the gas across the porous surface and the impact Weber number of the drop. The most remarkable aspect of the dynamics is the transformation of a drop into a bubble, which occurs when a drop just overcomes the repulsion by the gas flow and wets the surface. The transition to the regime in which a drop is transformed to a bubble is well described by a simple scaling relationship based on a balance between inertia and the repulsive force due to the gas flow.

This study explores the impact of small-scale variability in the bottom relief on the dynamics and evolution of broad baroclinic flows in the ocean. The analytical model presented here generalizes the previously reported barotropic ‘sandpaper’ theory of flow–topography interaction to density-stratified systems. The multiscale asymptotic analysis leads to an explicit representation of the large-scale effects of irregular bottom roughness. The utility of the multiscale model is demonstrated by applying it to the problem of topography-induced spin-down of an axisymmetric vortex. We find that bathymetry affects vortices by suppressing circulation in their deep regions. As a result, vortices located above rough topography tend to be more stable than their flat-bottom counterparts. The multiscale theory is validated by comparing corresponding topography-resolving and parametric simulations.

This study is an attempt at solving steady-state triple-deck equations using the method of Bos & Ruban (Phil. Trans. R. Soc. Lond. A, vol. 358, issue 1777, 2000, pp. 3063–3073) for supersonic flow past a compression corner. This was motivated by the fact that in their above paper, they show solutions for scale angles up to 8, the highest obtained so far in the literature. However, we encountered a stationary wave-packet at the corner for scale angles 1.82 and 1.96, depending on the values of stretching factors. Our solutions are then compared with the steady-state solutions produced using the method of Logue, Gajjar & Ruban (Phil. Trans. R. Soc. A, vol. 372, issue 2020, 2014, 20130342), which do not show such wave-packets. These wave-packets do not appear to be the result of flow instability, as flow instabilities should only appear with unsteady equations (Cassel, Ruban & Walker, J. Fluid Mech., vol. 300, 1995, pp. 265–285). It is therefore suggested that the method of Bos & Ruban (Phil. Trans. R. Soc. Lond. A, vol. 358, issue 1777, 2000, pp. 3063–3073) produces these spurious wave-packets as a consequence of their numerical method. This has important implications in the interpretation of triple-deck solutions.

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.

A liquid metal flow in the form of a submerged round jet entering a square duct in the presence of a transverse magnetic field is studied experimentally. A range of high Reynolds and Hartmann numbers is considered. Flow velocity is measured using electric potential difference probes. A detailed study of the flow in the duct's cross-section about seven jet's diameters downstream of the inlet reveals the dynamics, which is unsteady and dominated by high-amplitude fluctuations resulting from the instability of the jet. The flow structure and fluctuation properties are largely determined by the value of the Stuart number ${{N}}$. At moderate ${{N}}$, the mean velocity profile retains a central jet with three-dimensional perturbations increasingly suppressed by the magnetic field as ${{N}}$ grows. At higher values of ${{N}}$, the flow becomes quasi-two-dimensional and acquires the form of an asymmetric macrovortex, with high-amplitude velocity fluctuations reemerging.

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.

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.

Electrohydrodynamic sedimentation of simple drops has been a topic well-studied by researchers. However, electrohydrodynamic sedimentation of a compound drop would be critically influenced by the density of the involved phases and this has hitherto remained unaddressed. Herein, we develop a semi-analytical model for an eccentric compound drop settling under the action of gravity and an electric field using bispherical coordinates. The sedimentation velocity of the two drops (shell and core) is determined, and the same is applied to capture the influence of concomitant physical, hydrodynamic and electric properties on compound droplet sedimentation. The findings indicate that the compound drop may either sediment or de-sediment depending on the amount of eccentricity and its interplay with electrohydrodynamic parameters. Thereafter, the critical limit of eccentricity and time within which similar results are furnished by concentric and eccentric configurations is determined. It is found that, based on the property ratios, the eccentricity remains lower than 0.1 up to a non-dimensional time range of the order of 102–103, within which both the configurations can furnish a similar solution.

We study numerically a sequence of eddies in two-dimensional electrohydrodynamics (EHD) flows of a dielectric liquid, driven by an electric potential difference between a hyperbolic blade electrode and a flat plate electrode (or the blade–plate configuration). The electrically driven flow impinges on the plate to generate vortices, which resemble Moffatt eddies (Moffatt, J. Fluid Mech., vol. 18, 1964, pp. 1–18). Such a phenomenon in EHD was first reported in the experimental work of Perri et al. (J. Fluid Mech., vol. 900, 2020, A12). We conduct direct numerical simulations of the EHD flow with three Moffatt-type eddies in a large computational domain at moderate electric Rayleigh numbers ($T$, quantifying the strength of the electric field). The ratios of size and intensity of the adjacent eddies are examined, and they can be compared favourably to the theoretical prediction of Moffatt; interestingly, the quantitative comparison is remarkably accurate for the two eddies in the far field. Our investigation also shows that a larger $T$ strengthens the vortex intensity, and a stronger charge diffusion effect enlarges the vortex size. A sufficiently large $T$ can further result in an oscillating flow, consistent with the experimental observation. In addition, a global stability analysis of the steady blade–plate EHD flow is conducted. The global mode is characterised in detail at different values of $T$. When $T$ is large, the confinement effect of the geometry in the centre region may lead to an increased oscillation frequency. This work contributes to the quantitative characterisation of the Moffatt-type eddies in EHD flows.

Richtmyer–Meshkov instability induced by two successive shock waves is experimentally studied in a specific shock tube. To create two successive shock waves synchronously, a driver section is added between the driver and driven sections of the standard shock tube, and an electronically controlled membrane rupture equipment is adopted. The shock-tube flow after the membranes rupture is well described by combining the shock relations, isentropic wave relations with compatibility relations across the contact surface (region). The new shock tube is capable of generating two successive shock waves with controllable strengths and time interval, and provides a relatively ‘clean’ wave system. Then the developments of single-mode light–heavy interfaces with different initial conditions induced by two successive shock waves are investigated. The initial amplitudes are all small enough such that the first-shocked interface is within the linear growth regime at the arrival of the second shock. The results show that if the pre-second-shock perturbation amplitude is small, the linear, nonlinear and modal evolutions of the double-shocked interface can be reasonably predicted by the existing models proposed for predicting the perturbation growth induced by a single shock. For the double-shocked interface, the second shock provides an additional perturbation velocity field to the original one introduced by the first shock impact. The validity of the linear superposition model indicates that the linear superposition of these two perturbation velocity fields is satisfied. Therefore, a double-shocked interface evolves similarly to a single-shocked interface provided that their postshock amplitudes and linear growth rates are the same.

This paper presents a systematic analysis of direct and adjoint problems for sound propagation with flow. Two scalar propagation operators are considered: the linearised potential equation from Goldstein, and Pierce's equation based on a high-frequency approximation. For both models, the analysis involves compressible base flows, volume sources and surfaces that can be vibrating and/or acoustically lined (using the Myers impedance condition), as well as far-field radiation boundaries. For both models, the direct problems are fully described and adjoint problems are formulated to define tailored Green's functions. These Green's functions are devised to provide an explicit link between the direct problem solutions and the source terms. These adjoint problems and tailored Green's functions are particularly useful and efficient for source localisation problems, or when stochastic distributed sources are involved. The present analysis yields a number of new results, including the adjoint Myers condition for the linearised potential equation, as well as the formulation of the direct and adjoint Myers condition for Pierce's equation. It is also shown how the adjoint problems can be recast in forms that are readily solved using existing simulation tools for the direct problems. Results presented in this paper are obtained using a high-order finite element method. Several test cases serve as validation for the approach using tailored Green's functions. They also illustrate the relative benefits of the two propagation operators.

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.