The electrohydrodynamic processes taking place in a cone jet cause ohmic and viscous dissipation, and ultimately self-heating of the liquid. Despite this, previous analyses have modelled cone jets as isothermal systems. To investigate the validity of this assumption, this work applies the leaky-dielectric model to cone jets, while also requiring conservation of energy to reproduce the variation of temperature caused by dissipation and temperature-dependent liquid properties. The main goals are to determine whether there exist electrospraying conditions for which the isothermal assumption is inaccurate, and quantify the temperature field under such conditions. The work confirms that self-heating and thermal effects are important in liquids with sufficiently high conductivities, which is a significant limit because these electrical conductivities are needed to produce jets and droplets with radii of tens of nanometres or smaller. The numerical solution provides accurate expressions for evaluating the dissipation and the temperature increase in cone jets, and confirms that thermal effects cause the apparent breakdown of the traditional scaling law for the current of cone jets of highly conducting liquids.

In magnetoconvection, the flow of an electromagnetically conductive fluid is driven by a combination of buoyancy forces, which create the fluid motion due to thermal expansion and contraction, and Lorentz forces, which distort the convective flow structure in the presence of a magnetic field. The differences in the global flow structures in the buoyancy-dominated and Lorentz-force-dominated regimes lead to different heat transport properties in these regimes, reflected in distinct dimensionless scaling relations of the global heat flux (Nusselt number $Nu$) versus the strength of buoyancy (Rayleigh number $Ra$) and electromagnetic forces (Hartmann number $Ha$). Here, we propose a theoretical model for the transition between these two regimes for the case of a static vertical magnetic field applied across a convective fluid layer confined between two isothermal, a lower warmer and an upper colder, horizontal surfaces. The model suggests that the scaling exponents $\gamma$ in the buoyancy-dominated regime, $Nu\sim Ra ^\gamma$, and $\xi$ in the Lorentz-force-dominated regime, $Nu\sim (Ha^{-2}Ra)^\xi$, are related as $\xi =\gamma /(1-2\gamma )$, and the onset of the transition scales with $Ha^{-1/\gamma }Ra$. These theoretical results are supported by our direct numerical simulations for $10\leq Ha\leq 2000$, Prandtl number $Pr=0.025$ and $Ra$ up to $10^9$ and data from the literature.

The dipole–multipole transition in rapidly rotating dynamos is investigated through the analysis of forced magnetohydrodynamic waves in an unstably stratified fluid. The focus of this study is on the inertia-free limit applicable to planetary cores, where the Rossby number is small not only on the core depth but also on the length scale of columnar convection. By progressively increasing the buoyant forcing in a linear magnetoconvection model, the slow magnetic–Archimedean–Coriolis (MAC) waves are significantly attenuated so that their kinetic helicity decreases to zero; the fast MAC wave helicity, on the other hand, is practically unaffected. In turn, polarity reversals in low-inertia spherical dynamos are shown to occur when the slow MAC waves disappear under strong forcing. Two dynamically similar regimes are identified – the suppression of slow waves in a strongly forced dynamo and the excitation of slow waves in a moderately forced dynamo starting from a small seed field. While the former regime results in polarity reversals, the latter regime produces the axial dipole from a chaotic multipolar state. For either polarity transition, a local Rayleigh number based on the mean wavenumber of the energy-containing scales bears the same linear relationship with the square of the peak magnetic field measured at the transition. The self-similarity of the dipole–multipole transition can place a constraint on the Rayleigh number for polarity reversals in the Earth.

Three different porous substrates (with different pore sizes, $s$, and permeabilities, $K$) are used to examine their effect on the structure of boundary layer flow over them. The flow is characterised with single-point hot-wire measurements as well as planar particle image velocimetry. In order to elucidate differences in shallow and deep flows past porous substrates, foams with two different thicknesses ($h$) are used (for all three substrates). A wide range of friction Reynolds numbers ($2000< Re_\tau <13\,500$) and permeability-based Reynolds numbers ($1< Re_K< 50$) are attained. For substrates with $Re_K \sim 1$, the flow behaviour remains similar to flow over impermeable smooth walls and as such Townsend's hypothesis remains valid. Very large-scale motions are observed over permeable foams even when the $Re_K > 1$. In contrast, a substantial reduction in velocity disturbances and associated length scales is achieved for permeable foams with intermediate values of pore density and relative foam thickness ($h/s$), which affects outer-layer similarity. As permeability is increased by increasing pore size, the foam becomes sparse relative to viscous scales at high Reynolds numbers. For such foams, the flow conforms to outer-layer similarity and is more akin to flow over rough surfaces. Permeability attenuates the wavelengths associated with the outer-layer peak.

A collection of microswimmers immersed in an incompressible fluid is characterised by strong interactions due to the long-range nature of the hydrodynamic fields generated by individual organisms. As a result, suspensions of rear-actuated ‘pusher’ swimmers such as bacteria exhibit a collective motion state often referred to as ‘bacterial turbulence’, characterised by large-scale chaotic flows. The onset of collective motion in pusher suspensions is classically understood within the framework of mean-field kinetic theories for dipolar swimmers. In bulk two and three dimensions, the theory predicts that the instability leading to bacterial turbulence is due to mutual swimmer reorientation and sets in at the largest length scale available to the suspension. Here, we construct a similar kinetic theory for the case of a dipolar microswimmer suspension restricted to a two-dimensional plane embedded in a three-dimensional incompressible fluid. This setting qualitatively mimics the effect of swimming close to a two-dimensional interface. We show that the in-plane flow fields are effectively compressible in spite of the incompressibility of the three-dimensional bulk fluid, and that microswimmers on average act as sources (pushers) or sinks (pullers). We analyse the stability of the homogeneous and isotropic state, and find two types of instability that are qualitatively different from the bulk, three-dimensional case: first, we show that the analogue of the orientational pusher instability leading to bacterial turbulence in bulk systems instead occurs at the smallest length scale available to the system. Second, an instability associated with density variations arises in puller suspensions as a generic consequence of the effective in-plane compressibility. Given these qualitative differences with respect to the standard bulk setting, we conclude that confinement can have a crucial role in determining the collective behaviour of microswimmer suspensions.

In sandstorms and thunderclouds, turbulence-induced collisions between solid particles and ice crystals lead to inevitable triboelectrification. The charge segregation is usually size dependent, with small particles charged negatively and large particles charged positively. In this work, we perform numerical simulations to study the influence of charge segregation on the dynamics of bidispersed inertial particles in turbulence. Direct numerical simulations of homogeneous isotropic turbulence are performed with the Taylor Reynolds number ${Re}_{\lambda }=147.5$, while particles are subjected to both electrostatic interactions and fluid drag, with Stokes numbers of 1 and 10 for small and large particles, respectively. Coulomb repulsion/attraction is shown to effectively inhibit/enhance particle clustering within a short range. Besides, the mean relative velocity between same-size particles is found to rise as the particle charge increases because of the exclusion of low-velocity pairs, while the relative velocity between different-size particles is almost unaffected, emphasizing the dominant roles of differential inertia. The mean Coulomb-turbulence parameter, ${Ct}_0$, is then defined to characterize the competition between the Coulomb potential energy and the mean relative kinetic energy. In addition, a model is proposed to quantify the rate at which charged particles approach each other and to capture the transition of the particle relative motion from the turbulence-dominated regime to the electrostatic-dominated regime. Finally, the probability distribution function of the approach rate between particle pairs is examined, and its dependence on the Coulomb force is further discussed using the extended Coulomb-turbulence parameter.

The Zakharov equation describes the evolution of weakly nonlinear surface gravity waves for arbitrary spectral shape. For deep-water waves, results from the Zakharov equation are well established. However, for two-dimensional propagation, in intermediate and shallow water, there are problems related to the treatment of apparent singularities in the contribution of the wave-induced set-up to the evolution of the surface gravity waves. More specifically, the kernel in the integral term is characterized by a regular and an apparent singular contribution. Here, we show that the Davey–Stewartson equation can be directly derived from the Zakharov equation, also in the shallow water limit. This result provides guidance on how to treat the singular contribution to the evolution of the action variable. A relevant result that is obtained is that the growth rate obtained from the stability analysis of a plane wave in shallow water does not depend on the singular part of the kernel of the Zakharov equation.

The Reynolds number dependent flow resistance of heterogeneous rough surfaces is largely unknown at present. The present work provides novel reference data for spanwise-alternating sandpaper strips as one idealised case of a heterogeneous rough surface. Experimental data are presented and analysed in direct comparison with drag measurements of homogeneous sandpaper surfaces and numerical simulations. Based on the homogeneous roughness data, the related challenges and sensitivities for the evaluation of roughness functions from experiments and simulations are discussed. A hydraulic channel height is suggested as an alternative measure for the drag impact of rough surfaces in internal flows. For the investigated heterogeneous roughness, it is found that turbulent flow does not exhibit a fully rough flow behaviour, indicating that the assignment of an equivalent sand grain height as commonly applied for homogeneous roughness is not possible. A prediction of the drag behaviour of rough strips based on an average between rough and smooth drag curves appears promising, but requires further refinement to capture the impact of turbulent secondary flows and spatial transients linking smooth and rough surface parts. While turbulent secondary flow induced by the roughness strips yield significant spanwise variation of the mean velocity profile for the investigated rough strips, we show that the spanwise averaged velocity profiles collapse reasonably well with a smooth or homogeneous rough wall flow. This allows to extract a global roughness function from the spanwise averaged flow field in good agreement with the one deduced from global pressure drop measurements.

Floating objects will drift due to the action of surface gravity waves. This drift will depart from that of a perfect Lagrangian tracer due to both viscous effects (non-potential flow) and wave–body interaction (potential flow). We examine the drift of freely floating objects in regular (non-breaking) deep-water wave fields for object sizes that are large enough to cause significant diffraction. Systematic numerical simulations are performed using a hybrid numerical solver, qaleFOAM, which deals with both viscosity and wave–body interaction. For very small objects, the model predicts a wave-induced drift equal to the Stokes drift. For larger objects, the drift is generally greater and increases with object size (we examine object sizes up to $10\,\%$ of the wavelength). The effects of different shapes, sizes and submergence depths and steepnesses are examined. Furthermore, we derive a ‘diffraction-modified Stokes drift’ akin to Stokes (Trans. Camb. Phil. Soc., vol. 8, 1847, pp. 411–455), but based on the combination of incident, diffracted and radiated wave fields, which are based on potential-flow theory and obtained using the boundary element method. This diffraction-modified Stokes drift explains both qualitatively and quantitatively the increase in drift. Generally, round objects do not diffract the wave field significantly and do not experience a significant drift enhancement as a result. For box-shape objects, drift enhancement is greater for larger objects with greater submergence depths (we report an increase of $92\,\%$ for simulations without viscosity and $113\,\%$ with viscosity for a round-cornered box whose size is $10\,\%$ of the wavelength). We identify the specific standing wave pattern that arises near the object because of diffraction as the main cause of the enhanced drift. Viscosity plays a small positive role in the enhanced drift behaviour of large objects, increasing the drift further by approximately $20\,\%$.

The free-surface channel flow around a square cylinder is analysed, over a wide range of blocking ratios, using three-dimensional simulations. The state of the flow is characterised in terms of the Froude number upstream and downstream of the square cylinder. The simulations confirm the presence of the subcritical and choked states, and provide new insight into the supercritical state and band-gap through an analysis of how the momentum flux varies with Froude number along the channel. The influence of the blocking ratio on the flow state and drag force is analysed and shows the significant rise of drag in the choked regime.

In numerical studies of thermal convection that includes a layer of lighter surface fluid, the light fluid naturally forms clusters that bulge downward at downwelling sites. A curious result is that in some cases, the clusters have maximum bulging downward near the sides of the cluster instead of a single bulge downward centred above the downwelling. The fluid mechanics leading to this ‘double bulge’ formation is analysed. To accomplish this, a simplified model replaces the thermally driven convection cells with driving cells with a fixed speed. Adding a layer of dense fluid on the bottom to the previous configuration leads to bulges along the top and bottom. More importantly, this allows a new scaling that reduces the number of governing parameters from four to three and even to two in this study. The mechanism for the double bulges comes from buoyancy of the clusters. This produces localized vorticity at the sides of the cluster that has the opposite sign of the driving cells. When this vorticity is approximately the same order of magnitude as the driving cell vorticity, a divergence in the middle of each cluster leads to the double bulges. The effect can be so great that the underlying flow cells are tilted so that vertical motion is reversed under the middle of each bulge.

Recent advances in manufacturing techniques are providing porous media with both high permeability, necessary for effective passive flow control, and high structural strength, essential for engineering applications. We therefore examine the predictive accuracy of the standard Darcy–Forchheimer (DF) law, which is often used to model porous media flows, for inclusion Reynolds numbers (Re) ranging from the low linear regime to the high nonlinear regime where unsteady effects such as vortex shedding become evident. We consider two different inclusion shapes, square and circular, and three different arrangements of the inclusions – inline, staggered and random. The numerical simulations show that the DF law performs well for low-Re flows, irrespective of the inclusion configuration. For intermediate/high-Re flows, the DF law is adequate only when the arrangement is highly random. However, for the regularly arranged topologies or less random geometries at intermediate/high-Re flows, the DF-law performance diminishes significantly due to flow sheltering and redirection (‘inertial flow deflection’) effects that arise from flow inertia, separation and vortex shedding in the wake of the inclusions. It is shown that the standard DF law, in which the nonlinear permeability tensor is independent of orientation, does not capture such effects. We modified the DF law to capture flow redirection effects by allowing the Forchheimer permeability tensor to depend on the flow orientation with respect to the principal geometrical directions of the porous geometry, and examined this extended DF law for these flows.

A theoretical and computational study analysing the initiation of yield-stress fluid percolation in porous media is presented. Yield-stress fluid flows through porous media are complicated due to the nonlinear rheological behaviour of this type of fluid, rendering the conventional Darcy type approach invalid. A critical pressure gradient must be exceeded to commence the flow of a yield-stress fluid in a porous medium. As the first step in generalising the Darcy law for yield-stress fluids, a universal scale based on the variational formulation of the energy equation is derived for the critical pressure gradient which reduces to the purely geometrical feature of the porous media. The presented scaling is then validated by both exhaustive numerical simulations (using an adaptive finite element approach based on the augmented Lagrangian method), and also the previously published data. The considered porous media are constructed by randomised obstacles with various topologies; namely square, circular and alternatively polygonal obstacles which are mimicked based on Voronoi tessellation of circular cases. Moreover, computations for the bidispersed obstacle cases are performed which further demonstrate the validity of the proposed universal scaling.

A numerical investigation was conducted on $Re_{\varGamma _{0}}=3000$ vortex rings colliding with wall-mounted hemispheres to study how their relative sizes affect the resulting vortex dynamics and structures. The hemisphere to vortex ring diameter ratio ranges from $D/d=0.5$ to $D/d=2$. Secondary/tertiary vortex rings are observed to result from hemispheric surface boundary layer separations rather than wall boundary layer separations as the diameter ratio increases. While those for $D/d\leq 1$ hemispheres can be attributed to sequential hemispheric and wall boundary layer separations, the primary vortex ring produces a series of secondary/tertiary vortex rings only along the $D/d=2$ hemispheric surface. This indicates that the presence of the wall makes little difference when the hemisphere is sufficiently large. On top of comparing vortex ring circulations and translational velocities between hemisphere and flat-wall based collisions, present collision outcomes have also been compared with those predicted by specific discharge velocity models. Additionally, comparisons of vortex core trajectories and vortex ring formation locations with earlier cylindrical convex surface based collisions provide more clarity on differences between two- and three-dimensional convex surfaces. Finally, vortex flow models are presented to account for the significantly different flow behaviour as the hemisphere size varies. Specifically, the vortex flow model for the $D/d=2$ hemisphere hypothesizes that the recurring tertiary vortex ring formations cease only when the primary vortex ring slows down sufficiently for the last tertiary vortex ring to entangle with it and render it incoherent. Until that happens, the primary vortex ring will continue to induce more tertiary vortex rings to form, with potential implications for heat/mass transfer optimizations.

Simultaneous time-resolved measurements of wall deformation and the 3-D velocity field in boundary layers over a compliant surface are performed by integrating Mach Zehnder interferometry with tomographic particle tracking velocimetry. The pressure is calculated by spatially integrating the material acceleration. Combining data obtained from several references, trends of the deformation r.m.s. scaled by the compliant wall thickness collapse when plotted vs pressure fluctuations scaled by the material shear modulus. For the present data, at all Reynolds numbers, the deformation waves travel at 53% of the free-stream velocity and have a preferred wavelength of three times the thickness. The latter is consistent with theoretical models. Adopting insight derived from atmospheric wind–wave interactions, the pressure–deformation correlations peak at or slightly above the ‘critical layer’, where the mean flow speed is equal to the surface wave speed. This layer is located within the log layer, and when expressed using inner variables, increases in elevation with increasing Reynolds number. For the entire region below the critical layer, wavenumber–frequency spectra of pressure and vertical velocity fluctuations indicate that the turbulence is phase locked and travels with the deformation, even for deformation amplitudes much smaller than a wall unit. In contrast, above the critical layer, the turbulence is advected at the local mean streamwise velocity, and its correlation with the deformation decays rapidly. These findings indicate that the height of the zone dominated by flow-deformation interactions is determined by the surface wave speed, and its variations are caused by deformation-induced modifications to the mean velocity profile.

The first stages of the path instability phenomenon affecting the buoyancy-driven motion of gas bubbles rising in weakly or moderately viscous liquids are examined using a recently developed numerical approach designed to assess the global linear stability of incompressible flows involving freely evolving interfaces. Predictions for the critical bubble size and frequency of the most unstable mode are found to agree well with reference data obtained in ultrapure water and in several silicone oils. By varying the bubble size, stability diagrams are built for several specific fluids, revealing three distinct regimes with different bifurcation sequences. The spatial structure of the unstable modes is analysed, together with the variations of the bubble shape, position and orientation. For this purpose, displacements of the bubble surface are split into rigid-body components and volume-preserving deformations, allowing us to determine how the relative magnitude of the latter varies with the fluid properties and bubble size. Predictions obtained with freely deformable bubbles are compared with those found by maintaining the bubble shape determined in the base state frozen during the stability analysis. This comparison reveals that deformations leave the phenomenology of the first bifurcations unchanged in low-viscosity fluids, especially water. Hence, in such fluids, bubbles behave essentially as freely moving rigid bodies submitted to constant-force and zero-torque constraints, at the surface of which the fluid obeys a shear-free condition. In contrast, deformations change the nature of the primary bifurcation in oils slightly more viscous than water, whereas, somewhat surprisingly, they leave the near-threshold phenomenology unchanged in more viscous oils.

The secondary motion caused by turbulence anisotropy is one of the crucial factors for determining the size of corner-flow separation in a side-wall interference flow field. Therefore, through a wall-resolved large-eddy simulation (LES) of a side-wall interference flow field, this study investigates the effects of the secondary motion on the corner-flow separation and explores the turbulence modelling that can reproduce the secondary flow motion. The momentum transport analysis using the LES results shows that the secondary vortex has twofold effects on delaying the corner-flow separation: the convective transport of the streamwise momentum towards the corner, and the enhanced production of turbulence by increasing the shear. Also, the vorticity transport analysis reconfirms that the secondary motion is caused primarily by turbulence anisotropy in the outer layer of the turbulent boundary layer. Furthermore, a quadratic constitutive relation (QCR) is proposed based on the analysis of the relationship between the Reynolds stress and velocity gradient. The proposed QCR consists of two quadratic terms and three constant parameters. The a priori analysis using the LES data shows that the proposed QCR represents the anisotropy of the Reynolds stress overall better than the existing QCR. Reynolds-averaged Navier–Stokes simulation using the proposed QCR with the Spalart–Allmaras turbulence model shows improvements in the prediction of the corner-flow separation compared to the results obtained by the existing QCR with the same turbulence model.

A data-driven implementation of a quasi-linear approximation is presented, extending a minimal quasi-linear approximation (MQLA) (Hwang & Ekchardt, J. Fluid Mech., vol. 894, 2020, p. A23) to incorporate non-zero streamwise Fourier modes. A data-based approach is proposed, matching the two-dimensional wavenumber spectra for a fixed spanwise wavenumber between a direct numerical simulation (DNS) (Lee & Moser, J. Fluid Mech., vol. 774, 2015, pp. 395–415) and that generated by the eddy viscosity enhanced linearised Navier–Stokes equations at $Re_\tau \approx 5200$, where $Re_\tau$ is the friction Reynolds number. Leveraging the self-similar nature of the energy-containing part in the DNS velocity spectra, a universal self-similar streamwise wavenumber weight is determined for the linearised fluctuation equations at $Re_\tau \simeq ~5200$. The data-driven quasi-linear approximation (DQLA) provides noteworthy enhancements in the wall-normal and spanwise turbulence intensity profiles. It exhibits a qualitatively similar structure in the spanwise wavenumber velocity spectra compared with the MQLA. Additionally, the DQLA offers extra statistical outputs in the streamwise wavenumber coordinates, enabling a comprehensive global analysis of this modelling approach. By comparing the DQLA results with DNS results, the limitations of the presented framework are discussed, mainly pertaining to the lack of the streak instability (or transient growth) mechanism and energy cascade from the linearised model. The DQLA is subsequently employed over a range of Reynolds numbers up to $Re_\tau = 10^5$. Overall, the turbulence statistics and spectra produced by the DQLA scale consistently with the available DNS and experimental data, with the Townsend–Perry constants displaying a mild Reynolds dependence (Hwang, Hutchins & Marusic, J. Fluid Mech., vol. 933, 2022, p. A8). The scaling behaviour of the turbulence intensity profiles deviates away from the classic $\ln (Re_\tau )$ scaling, following the inverse centreline velocity scaling for the higher Reynolds numbers.

Large-amplitude gust encounters exhibit a range of separated flow phenomena, making them difficult to characterize using the traditional tools of aerodynamics. In this work, we propose a dynamical systems approach to gust encounters, viewing the flow as a cycle (or a closed trajectory) in state space. We posit that the topology of this cycle, or its shape and structure, provides a compact description of the flow, and can be used to identify coordinates in which the dynamics evolve in a simple, intuitive way. To demonstrate this idea, we consider flowfield measurements of a transverse gust encounter. For each case in the dataset, we characterize the full-state dynamics of the flow using persistent homology, a tool that identifies holes in point cloud data, and transform the dynamics to a reduced-order space using a nonlinear autoencoder. Critically, we constrain the autoencoder such that it preserves topologically relevant features of the original dynamics, or those features identified by persistent homology. Using this approach, we are able to transform six separate gust encounters to a three-dimensional latent space, in which each gust encounter reduces to a simple circle, and from which the original flow can be reconstructed. This result shows that topology can guide the creation of low-dimensional state representations for strong transverse gust encounters, a crucial step towards the modelling and control of aerofoil–gust interactions.

A fundamental open problem in the mechanics of granular media is the determination of the stress in the static state. It is known that the static stress depends strongly on how the grain assembly is created and the nature of confining boundaries. Non-trivial spatial variations have been observed even in simple geometries, posing long-standing challenges to continuum modelling. In this paper, we create gravity-deposited grain packings computationally and devise a method to visualise the paths of load transmission, which we call force lines. We show that the force lines reflect the flow during deposition, thereby encoding preparation history. We then show that the force lines coincide with ensemble averaged biased random walks in the particle contact network; this identification yields a closure relation for the stress, which together with the static momentum balances fully determines the stress field. The model makes accurate predictions for the stress in piles and silos, even for unusual deposition methods, thereby showing promise for more general scenarios.