As most mathematically justifiable Lagrangian coherent structure detection methods rely on spatial derivatives, their applicability to sparse trajectory data has been limited. For experimental fluid dynamicists and natural scientists working with Lagrangian trajectory data via passive tracers in unsteady flows (e.g. Lagrangian particle tracking or ocean buoys), obtaining material measures of fluid rotation or stretching is an active topic of research. To facilitate frame-indifferent investigations in unsteady and sparsely sampled flows, we present a novel approach to quantify fluid stretching and rotation via relative Lagrangian velocities. This technique provides a formal objective extension of quasi-objective metrics to unsteady flows by accounting for mean flow behaviour. For extremely sparse experimental data, fluid structures may be significantly undersampled and the mean flow behaviour becomes difficult to quantify. We provide a means to maintain the accuracy of our novel sparse flow diagnostics in extremely sparse sampling scenarios, such as ocean buoy data and Lagrangian particle tracking. We use data from multiple numerical and experimental flows to show that our methods can identify structures beyond existing limits of sparse, frame-indifferent diagnostics and exhibit improved interpretability over common frame-dependent diagnostics.

The flow near a moving contact line depends on the dynamic contact angle, viscosity ratio and capillary number. We report experiments involving immersing a plate into a liquid bath, concurrently measuring the interface shape, interfacial velocity and fluid flow using digital image processing and particle image velocimetry. All experiments were performed at low plate speeds to maintain small Reynolds and capillary numbers for comparison with viscous theories. The dynamic contact angle, measured in the viscous phase, was kept below $90^{\circ }$ and the viscosity ratio, $\lambda < 1$. This region of parameter space is largely unexplored for advancing contact lines. An important aim of the present study is to provide new experimental data against which new contact line models can be developed. The flow field is directly compared against the prediction from the viscous theory of Huh & Scriven (J. Colloid Interface Sci., vol. 35, issue 1, 1971, pp. 85–101) but with a slight modification involving the curved interface. Remarkable agreement is found between experiments and theory across a wide parameter range. The prediction for interfacial speed from Huh & Scriven is also in excellent agreement with experiments except in the vicinity of the contact line. Material points along the interface were found to rapidly slow down near the contact line, thus alleviating the singularity at the moving contact line. To the best of our knowledge, such a detailed test of theoretical models has not been performed before and we hope the present study will spur new modelling efforts in the field.

The spontaneous formation of zonal jets is a distinctive feature of geostrophic turbulence with the phenomenon witnessed in numerous numerical studies. In such systems, strong rotation anisotropises the spectral evolution of the energy density such that zonal modes are favoured. In physical space, this manifests as eddies zonally elongating and forming into zonal jets. In the presence of large scale dissipation, the flow may reach statistical stationarity such that the zonal structure persists in the zonal and time mean, and is supported by a flux of eddy momentum. What is unclear is how the excitation of Rossby waves arranges the underlying eddy momentum stresses to support the mean flow structures. To study this, we examine a steady-state flow in the so-called ‘zonostrophic’ regime, in which characteristic scales of geostrophic turbulence are well separated and there are several alternating zonal jets that have formed spontaneously. We apply a geometric eddy ellipse formulation, in which momentum fluxes are cast as ellipses that encode information about the magnitude and direction of flux; the latter is described using the tilt angle. With the aid of a zonal filter, it is revealed that the scales responsible for providing the momentum fluxes associated with the jet structure are much smaller than the characteristic scales identified, and occupy a region of the energy spectrum that has been typically associated with isotropic dynamics.

A novel experiment was performed in rotating Rayleigh–Bénard convection (RRBC), wherein the convection cell with radius $R$ was shifted away from the rotation axis by a distance $d$. In this case, the centrifugal force felt by a fluid parcel (characterized by the Froude number $Fr$) can be decomposed into an axisymmetrical part $Fr_R$ and a directed one $Fr_d$. It has been reported that the global heat transport enhances at $Fr_{d,c}$ and then reaches an optimal state at $Fr_{d,max}$ (Hu et al., Phys. Rev. Lett., vol. 127, 2021, 244501). In this paper, the local properties after the offset effects set in are investigated further, which show different features before and after $Fr_{d,max}$. The local temperature measurements at the cell centre reveal that the bulk flow turns from a turbulent state into a laminar state at $Fr_{d,max}$, which is consistent with the particle image velocimetry results. This transition can be qualitatively understood by an equivalent tilted RRBC system. As for the hot and cold coherent structures near the sidewall, their vertical temperature variations reach a minimum at $Fr_{d,max}$, implying that these structures are mostly uniform in the vertical direction at $Fr_{d,max}$. Their temperature contrasts show a linear dependence on $Fr_d$ and start to deviate from this linear behaviour when $Fr_d>Fr_{d,max}$. Besides the dominant effects of $Fr_d$, the secondary effects of $Fr_R$ are also investigated. Due to the positive effect of $+Fr_R$ on the cold structure and the negative effect of $-Fr_R$ on the hot one, the cold structure is more coherent than the hot one, but its size is smaller. The shift of the cold cluster centre from the farthest point is also larger than the shift of the hot one from the nearest point.

We use $123$ three-dimensional direct numerical simulations to study fingering convection in non-rotating spherical shells. We investigate the scaling behaviour of the flow length scale, the non-dimensional heat and compositional fluxes $Nu$ and $Sh$ and the mean convective velocity over the fingering convection instability domain defined by $1 \leq R_\rho < Le$, $R_\rho$ being the ratio of density perturbations of thermal and compositional origins and $Le$ the Lewis number. We show that the chemical boundary layers are marginally unstable and adhere to the laminar Prandtl–Blasius model, hence explaining the asymmetry between the inner and outer spherical shell boundary layers. We develop scaling laws for two asymptotic regimes close to the two edges of the instability domain, namely $R_\rho \lesssim Le$ and $R_\rho \gtrsim 1$. For the former, we develop novel power laws of a small parameter $\epsilon$ measuring the distance to onset, which differ from theoretical laws published to date in Cartesian geometry. For the latter, we find that the Sherwood number $Sh$ gradually approaches a scaling $Sh\sim Ra_\xi ^{1/3}$ when $Ra_\xi \gg 1$; and that the Péclet number accordingly follows $Pe \sim Ra_\xi ^{2/3} |Ra_T|^{-1/4}$, $Ra_T$ and $Ra_{\xi}$ being the thermal and chemical Rayleigh numbers. When the Reynolds number exceeds a few tens, we report on a secondary instability which takes the form of large-scale toroidal jets which span the entire spherical domain. Jets distort the fingers, resulting in Reynolds stress correlations, which in turn feed the jet growth until saturation. This nonlinear phenomenon can yield relaxation oscillation cycles.

Direct numerical simulation is performed for flow separation over a bump in a turbulent channel. Comparisons are made between a smooth bump and one where the lee side is covered with replicas of shark denticles – dermal scales that consist of a slender base (the neck) and a wide top (the crown). As flow over the bump is under an adverse pressure gradient (APG), a reverse pore flow is formed in the porous cavity region underneath the crowns of the denticle array. Remarkable thrust is generated by the reverse pore flow as denticle necks accelerate the fluid passing between them in the upstream direction. Several geometrical features of shark denticles, including some that had not previously been considered hydrodynamically functional, are identified to form the two-layer denticle structure that enables and sustains the reverse pore flow and thrust generation. The reverse pore flow is activated by the APG before massive flow detachment. The results indicate a proactive, on-demand drag reduction mechanism that leverages and transforms the APG into a favourable outcome.

Numerous flying and swimming creatures use the ground effect to boost their propulsive performance, with the ‘ground’ referring to either a solid boundary or a free surface. While our knowledge of how a solid boundary affects biolocomotion is relatively comprehensive, the ground effect of a free surface is not fully understood. To address this limitation, we conduct a numerical investigation on the propulsion performance of a flapping plate under a free surface, subject to a range of control parameters. When the Froude number ($Fr$) is very low (i.e. little surface deformation), the effects of a free surface are similar to those of a solid boundary, with enhanced thrust and input power but little change in efficiency. However, as $Fr$ increases (i.e. more surface deformation), our results reveal an optimal $Fr$ of approximately 0.6, where the free surface induces a more streamlined flow around the flapping plate, effectively reducing the added mass. This results in a significant decrease in input power and greatly enhanced efficiency.

Oscillatory flows have become an indispensable tool in microfluidics, inducing inertial effects for displacing and manipulating fluid-borne objects in a reliable, controllable and label-free fashion. However, the quantitative description of such effects has been confined to limit cases and specialized scenarios. Here we develop an analytical formalism yielding the equation of motion of density-mismatched spherical particles in oscillatory background flows, generalizing previous work. Inertial force terms are systematically derived from the geometry of the flow field together with analytically known Stokes number dependences. Supported by independent, first-principles direct numerical simulations, we find that these forces are important even for nearly density-matched objects such as cells or bacteria, enabling their fast displacement and separation. Our formalism thus consistently incorporates particle inertia into the Maxey–Riley equation, and in doing so provides a generalization of Auton's modification to added mass, as well as recovering the description of acoustic radiation forces on particles as a limiting case.

Although the interaction between microswimmers and walls during near-wall swimming has been extensively studied, the effect of microswimmer shapes and slip boundary conditions on the dynamic characteristics of near-wall microswimmers has received less attention. In this study, elliptical microswimmer models have been developed with various aspect ratios based on circular microswimmers. The lattice Boltzmann method has been used for the numerical simulation of the dynamic behaviour of microswimmers near walls. Under slip boundary conditions, the escape or capture of microswimmers by the walls is influenced by the swimming Reynolds number (Res), wall slip length (ls) and the aspect ratio (Cab) of a microswimmer. Changes in the Cab value of a microswimmer considerably affect its swimming state, especially for puller-type microswimmers. The tendency of pullers to be captured by the wall increases with increasing Cab. Moreover, changes in ls within the slip boundary condition of a puller can induce a transition in its movement state from a wall oscillation state to a stable sliding state and eventually to a wall lock-up state, a process influenced by the Cab value of the puller. Pusher-type microswimmers show a considerably increased tendency to escape from walls with increasing Cab and no wall lock-up state is observed, which is opposite to the case of pullers. Pushers and pullers show an increased tendency to be captured by the wall with increasing initial swimming angle of the microswimmer. The findings of this study enhance our understanding of the swimming patterns of natural microswimmers near walls and are of substantial importance for the design of artificial microswimmers and microfluidic devices.

This study conducts particle-resolved direct numerical simulations to analyse how finite-size spherical particles affect the decay rate of turbulent kinetic energy in non-sustained homogeneous isotropic turbulence. The decaying particle-laden homogeneous isotropic turbulence is generated with two set-ups, i.e. (1) releasing particles into a single-phase decaying homogeneous isotropic turbulence and (2) switching off the driving force of a sustained particle-laden homogeneous isotropic turbulence. With both set-ups, the decay of turbulent kinetic energy follows a power-law when the flow is fully relaxed, similar to their single-phase counterparts. The dependence of the power-law decay exponent $n$ on the particle-to-fluid density ratio, particle size and volume fraction is also investigated, and a predictive model is developed. We find that the presence of heavier particles slows down the long-time power-law decay exponent.

We investigate the spreading of falling ambient-temperature Newtonian drops after their normal impact on a quartz plate covered with a thin layer of liquid nitrogen. As a drop expands, liquid nitrogen evaporates, generating a vapour film that maintains the drop in levitation. Consequently, the latter spreads in inverse Leidenfrost conditions. Three drop-spreading regimes are observed: (i) inertio-capillary, (ii) inertio-viscous, and (iii) inertio-viscous-capillary. In the first regime, although the drop expansion is essentially driven by a competition between inertial and capillary stresses, it is also affected by viscous effects emerging from the vapour film, which ultimately favours the development of a shear flow within the drop. Interestingly, vapour film effects become marginal in both the second and third regimes, allowing the drop to undergo biaxial extension primarily. More specifically, in the inertio-viscous scenario, the expansion is driven by the balance between inertial and biaxial extensional viscous stresses in the drop. Finally, inertia, capillarity and drop viscosity are all relevant in the third regime. These physical mechanisms are underlined through a mixed approach combining experiments with multiphase three-dimensional numerical simulations in light of spreading dynamics analyses, energy transfer and scaling laws. Our results are rationalized in a two-dimensional diagram linking the drops’ maximum expansion and spreading time with the observed spreading regimes through a single dimensionless parameter given by the square root of the capillary number (the ratio of the viscous stress to the capillary stress).

Transitional separating flow induced by a rectangular plate subjected to uniform incoming flow at Reynolds number (based on the incoming velocity and half plate height) 2000 is investigated using direct numerical simulation. The objective is to unveil the long-lasting mystery of low-frequency flapping motion (FM) in flow separation. At a fixed streamwise-vertical plane or from the perspective of previous experimental studies using pointwise or planar measurements, FM manifests as a low-frequency periodic switching between low and high velocities covering the entire separation bubble. The results indicate that in three-dimensional space, FM reflects an intricate evolution of streamwise elongated streaky structures under the influence of separated shear layer and mean flow reversal. The FM is an absolute instability, and is initiated through a lift-up mechanism boosted by mean flow deceleration near the crest of the separating streamline. At this particular location, the shear bends the vortex filament abruptly, so that one end is vertically struck into the first half of the separation bubble, whereas the other end is extended in the streamwise direction in the second half of the separation bubble. These two ends of vortex filament are mutually sustained and also stretched by the vertical acceleration and streamwise acceleration in the first and second halves of the separation bubble, respectively. This process periodically switches the low-velocity (or high-velocity) streaky structure to a high-velocity (or low-velocity) streaky structure encompassing the entire separation bubble, and thus flaps the separated shear layer up and down in the vertical direction. A ‘large vortex’ shedding manifests when the streaky structure switches signs. This large vortex is fundamentally different from the spanwise vortex shedding residing in the shear layer originated from the Kelvin–Helmholtz instability and successive vortex amalgamation. It is also believed that the three-dimensional evolution of streaky structures in the form of FM is applicable for both geometry- and pressure-induced separating flows.

Experiments are conducted to explore the rolling of a cylinder over a pool of viscous fluid. The speed, width and loading of the cylinder are varied along with the initial depth and length of the viscous pool. Depending on the conditions, the cylinder will either ride on a lubrication film or remain in solid contact with the underlying substrate. For the former situation, a lubrication theory is presented that describes the pressure underneath the cylinder and the thickness of the film. The theory approximates the flow by the one-dimensional Reynolds equation with the addition of one term, with an adjustable parameter, to account for the flux of fluid to the cylinder sides. Once this parameter is calibrated against experiment, the theory predicts peak lubrication pressures, gap sizes and film thicknesses to within approximately ten per cent. For lubricated rolling, the film splits evenly between the cylinder and substrate downstream of the nip. The printer's instability arises during the splitting process, patterning the residual fluid films on the substrate and cylinder. If the pool length is less than the cylinder circumference, the fluid adhering to the cylinder is rotated back into contact with the substrate, and when there is sufficient adhered fluid a lubrication film forms that can again be modelled by the theory. Conversely, if there is insufficient adhered fluid, no contiguous lubrication film is formed; instead, the pattern from the printer's instability ‘prints’ from the cylinder to the substrate.

Thermocapillary droplets with internal thermal singularities have potential applications in drug delivery and cell analysis. Inspired by the work of Pak et al. (J. Fluid Mech., vol. 753, 2014, pp. 535–552), which was investigated for a surfactant-laden non-deformable droplet in an isothermal Poiseuille flow, we have explored the droplet dynamics by taking account of additional internal thermal singularities, namely monopole and dipole. A generalized mathematical model is developed, which is solved by using the solenoidal decomposition to describe the flow field in any arbitrary Stokes flow, and results are shown extensively for the case of a non-isothermal Poiseuille flow. Under small Péclet number ($Pe_s$) limit, the droplet with an off-centred monopole or a dipole oriented along the flow direction shows cross-stream migration at $O(Pe_s^2)$. However, a dipole oriented perpendicular to the flow direction results in an $O(1)$ effect due to thermocapillarity, and from $O(Pe_s)$ onwards, we observe the combined impact of thermocapillary and surfactant-induced Marangoni stresses. As a surprise, we see cross-stream migration of the droplet from the Poiseuille flow centreline in a non-isothermal field, in contrast to existing findings which rule out any cross-stream migration. We show the trade-off between thermal Marangoni number ($Ma_T$) and surfactant Marangoni number ($Ma_\varGamma$). Our findings on droplet dynamics inspire new possibilities for microfluidics-based design.

When two fluid drops touch, they coalesce due to surface tension. At early times, there is only a relatively small fluid bridge joining the drops. An asymptotic solution is presented for an inertial regime of early-time coalescence, in which inertial forces balance surface tension at leading order. It is demonstrated that viscosity nevertheless has a leading-order effect. Radial momentum is created at the tightly curved edge of the fluid bridge by the net force $2\gamma$ (per unit length) due to surface tension. This momentum is left behind the radially expanding bridge edge in a thin viscous wake. The divergent volume flux in the wake entrains fluid from above and below the bridge, and drives an inviscid irrotational flow in the drops on the scale of the bridge radius. This flow widens the gap between the drops ahead of the bridge, and the larger gap width results in a lower rate of coalescence. Including viscosity in this way improves the agreement between theory and the available experimental and numerical data.

At high incidence, low-aspect-ratio wings present a unique set of aerodynamic characteristics, including flow separation, vortex shedding and unsteady force production. Furthermore, low-aspect-ratio wings exhibit a highly impactful tip vortex, which introduces strong spanwise gradients into an already complex flow. In this work, we explore the interaction between leading-edge flow separation and a strong, persistent tip vortex over a Reynolds number range of $600 \leq Re \leq 10{\,}000$. In performing this study, we aim to bridge the insight gained from existing low-Reynolds-number studies of separated flow on finite wings ($Re \approx 10^2$) and turbulent flows at higher Reynolds numbers ($Re \approx 10^4$). Our study suggests two primary effects of the Reynolds number. First, we observe a break from periodicity, along with a dramatic increase in the intensity and concentration of small-scale eddies, as we shift from $Re = 600$ to $Re = 2500$. Second, we observe that many of our flow diagnostics, including the time-averaged aerodynamic force, exhibit reduced sensitivity to Reynolds number beyond $Re = 2500$, an observation attributed to the stabilising impact of the wing tip vortex. This latter point illustrates the manner by which the tip vortex drives flow over low-aspect-ratio wings, and provides insight into how our existing understanding of this flow field may be adjusted for higher-Reynolds-number applications.

Time-resolved particle image velocimetry measurements were performed downstream of a swept backward-facing step. The measurements allow detailed analysis of the interactions between the unsteady instabilities and the stationary crossflow vortices. Different mechanisms are identified that lead to the modulation of the different families of unsteady instabilities that occur downstream of the step. For the low-frequency spanwise-travelling mode, the modulation occurs due to a redistribution of momentum when the instability encounters regions of large spanwise shear of the wall-normal and streamwise velocity. However, the higher-frequency streamwise-travelling instabilities undergo the familiar ‘lift-up’ mechanism when they encounter the regions of large vertical velocity due to the presence of the stationary crossflow vortices. The process leading to large velocity spikes, and ultimately to a laminar breakdown to turbulence, is identified as a constructive interaction between the different unsteady instabilities, coupled with an interaction with the stationary crossflow vortices when the phases align properly.

The hydrodynamic cavitation in semidilute solution flows of a flexible polymer additive in water was experimentally explored in a mesoscale converging–diverging nozzle to elucidate the cavitation reduction effects of polymer additives. Rheological measurements demonstrated that polymer solutions were shear-thinning, with infinite viscosities larger than pure water. The polymer additives significantly mitigated the intensity of cloud cavitation and the growth of violent cavity structures in the tested solution concentrations. Under conditions of supercavitation, the tested polymer solutions could not suppress the growth of large structures but showed a reduction in the population of cavitation bubbles. The temporal evolution and spatial variation of cavitation structures in different concentrations were captured using high-speed imaging. Statistical analysis of the images showed that polymers reduce cavitation via three main mechanisms. (1) The longitudinal expansion of cavities downstream is attenuated relative to the pure water. The streamwise distribution of vapour-ratio fluctuations was flattened, and its peak was shifted upstream in the solutions. (2) Mean collapse and growth rate of cavitating bubble pockets and their fluctuations were noticeably relaxed by polymer additives. For a 400 p.p.m. solution (parts per million (p.p.m.)), a reduction of 65 % was measured relative to pure water flow at the highest tested flow rate. (3) Spectral analysis of the downstream pressure indicated that the shedding frequency at the cavitation inception was reduced as the solution's concentration increased. This reduction was as high as 70 % for a 400 p.p.m. solution. These results highlight the strong interplay between polymer additives and the generation of cavitation-related structures.

We examined the influences of slip parameters on the velocity and thermal characteristics of a ferrofluid film of fixed thickness. The flow is generated on a rough and inclined whirling surface that is positioned in an external magnetic (dipole) field. The similarity transformation reduces the model equations (continuity, momentum, energy and concentration), and the solution of the normalized coupled ordinary differential equations is carried out through the finite element process. The influences of slip effects, Brownian motion, thermophoresis and a heat source on the velocity (radial, tangential and axial), gravity (drainage, induced), temperature profile and concentration profile are determined. The tangential flow and temperature are both decreased by an increase in the velocity slip parameter, whereas drainage, induced, radial and axial flows are increased. Enlarging the thermal slip parameter decreases the temperature. Improving slip parameters (velocity and thermal) also improves the concentration profile. Both Nusselt and Sherwood numbers are found to improve on improving the velocity slip parameter, while they decrease on decreasing the thermal slip parameter. The results and insights from this work could be applied to a wide range of medicinal fields, such as targeted medication therapy and delivery, tissue engineering, etc. as well as different industrial processes including coating, lubrication, heat transfer, etc.

We study a thin, laterally confined heated liquid layer subjected to mechanical parametric forcing without gravity. In the absence of parametric forcing, the liquid layer exhibits the Marangoni instability, provided the temperature difference across the layer exceeds a threshold. This threshold varies with the perturbation wavenumber, according to a curve with two minima, which correspond to long- and short-wave instability modes. The most unstable mode depends on the lateral confinement of the liquid layer. In wide containers, the long-wave mode is typically observed, and this can lead to the formation of dry spots. We focus on this mode, as the short-wave mode is found to be unaffected by parametric forcing. We use linear stability analysis and nonlinear computations based on a reduced-order model to investigate how parametric forcing can prevent the formation of dry spots. At low forcing frequencies, the liquid film can be rendered linearly stable within a finite range of forcing amplitudes, which decreases with increasing frequency and ultimately disappears at a cutoff frequency. Outside this range, the flow becomes unstable to either the Marangoni instability (for small amplitudes) or the Faraday instability (for large amplitudes). At high frequencies, beyond the cutoff frequency, linear stabilization through parametric forcing is not possible. However, a nonlinear saturation mechanism, occurring at forcing amplitudes below the Faraday instability threshold, can greatly reduce the film surface deformation and therefore prevent dry spots. Although dry spots can also be avoided at larger forcing amplitudes, this comes at the expense of generating large-amplitude Faraday waves.