In a theoretical analysis, we generalise well-known asymptotic results to obtain expressions for the rate of transfer of material from the surface of an arbitrary, rigid particle suspended in an open pathline flow at large Péclet number, ${Pe}$. The flow may be steady or periodic in time. We apply this result to numerically evaluate expressions for the surface flux to a freely suspended, axisymmetric ellipsoid (spheroid) in Stokes flow driven by a steady linear shear. We complement these analytical predictions with numerical simulations conducted over a range of ${Pe} = 10^1\text {--}10^4$ and confirm good agreement at large Péclet number. Our results allow us to examine the influence of particle shape upon the surface flux for a broad class of flows. When the background flow is irrotational, the surface flux is steady and is prescribed by three parameters only: the Péclet number, the particle aspect ratio and the strain topology. We observe that slender prolate spheroids tend to experience a higher surface flux compared to oblate spheroids with equivalent surface area. For rotational flows, particles may begin to spin or tumble, which may suppress or augment the convective transfer due to a realignment of the particle with respect to the strain field.

The breakup of droplets in a high-speed air stream is investigated experimentally and theoretically. This study is based on experiments conducted by exposing pendant droplets to a high-speed air jet, which allowed for imaging of the droplets at high temporal and spatial resolutions while maintaining a large field of view to capture the breakup process. Two factors of primary importance in droplet breakup are the breakup morphology and the resulting child droplet sizes. However, benchmark models have erroneous assumptions that impede the prediction of both morphology and breakup size and give non-physical geometries. The present theoretical work focuses on the ‘internal flow’ hypothesis, which suggests that the droplet's internal flow governs its breakup. The breakup process is subdivided into four stages: droplet deformation, rim formation, rim expansion and rim breakup. The internal flow mechanism is used to mathematically model the first two stages. The rim expansion is related to the growth of bags, while its breakup is shown to be due to Rayleigh–Plateau capillary instability. It is found that the breakup morphology is related to the division of the droplet into a disk that forms on the windward face and the undeformed droplet core. The amount of the droplet volume contained in the undeformed core decides the breakup morphology. Using this framework, it is shown that a three-step calculation can give an improved prediction of child drop sizes and morphology.

In karst and alpine areas, the interactions between water and rocks give rise to a large variety of marvellous patterns. In this work, we provide a hydrodynamic model for the formation of dissolutional patterns made of parallel longitudinal channels, commonly referred to as linear karren forms. The model addresses a laminar film of water flowing on a rock that is dissolving. The results show that a transverse instability of the water–rock system leads to a longitudinal channelization responsible for the pattern formation. The instability arises because of a positive feedback within the channels between the higher water flow and the enhanced chemical dissolution. The spatial scales predicted by the linear stability analysis span different orders of magnitude depending on the Reynolds number. This may explain why similar patterns of different sizes are observed on natural rocks. Results also show that the rock solubility affects just the temporal scale of the instability and the rock inclination plays a minor role in the pattern formation. It is eventually discussed how rain is not strictly necessary for the appearance of linear karren patterns, but it may affect some of their features.

Spanwise velocity statistics from high-Reynolds-number turbulent boundary layers are reported. The dataset combines efforts spanning over a decade at the University of Melbourne to accurately capture Reynolds number ($Re$) trends for the spanwise velocity, nominally over one order of magnitude change in $Re$, using custom subminiature cross-wire probes that minimise spatial resolution effects and misalignment errors. The spanwise velocity ($v$) variance is found to exhibit an $Re$ invariant logarithmic slope in the log region, in a similar manner to the streamwise velocity ($u$), which is consistent with the existence of self-similar features within wall-bounded flows. However, unlike the $u$-variance, it appears that the logarithmic $v$-variance trend continues to extend towards the wall. The increase in the $v$-variance with $Re$ in the log region is found to be due to ‘intermediate-scale eddies’, which follow distance-from-the-wall scaling. This results in the $v$-spectrogram exhibiting a dominant energetic ridge across the intermediate-scales, a trend that is not clearly observed in the $u$-spectrogram. Other features of the $v$-spectrogram are found to be similar to the $u$-spectrogram, such as showing small-scale near-wall features that scale universally with viscous units, and the influence of large-scale $v$ signals residing in the log region that extend to the wall, resulting in a large-scale $v$ footprint in the near-wall region. The observed behaviour of the $v$-spectrogram with changing $Re$ is used to construct a model for the $v$-variance based on contributions from small-, intermediate- and large-scales, leading to a predictive tool at asymptotically high $Re$.

Direct numerical simulations (DNS) of oscillatory flow around a cylinder show that the Stokes–Wang (S–W) solution agrees exceptionally well with DNS results over a much larger parameter space than the constraints of $\beta K^2\ll 1$ and $\beta \gg 1$ specified by the S–W solution, where $K$ is the Keulegan–Carpenter number and $\beta$ is the Stokes number. The ratio of drag coefficients predicted by DNS and the S–W solution, $\varLambda _K$, mapped out in the $K\text {--}\beta$ space, shows that $\varLambda _K < 1.05$ for $K\leq {\sim }0.8$ and $1 \leq \beta \leq 10^6$, which contradicts its counterpart based on experimental results. The large $\varLambda _K$ values are primarily induced by the flow separation on the cylinder surface, rather than the development of three-dimensional (Honji) instabilities. The difference between two-dimensional and three-dimensional DNS results is less than 2 % for $K$ smaller than the corresponding $K$ values on the iso-line of $\varLambda _K = 1.1$ with $\beta = 200\text {--}20\,950$. The flow separation actually occurs over the parameter space where $\varLambda _K\approx 1.0$. It is the spatio-temporal extent of flow separation rather than separation itself that causes large $\varLambda _K$ values. The proposed measure for the spatio-temporal extent, which is more sensitive to $K$ than $\beta$, correlates extremely well with $\varLambda _K$. The conventional Morison equation with a quadratic drag component is fundamentally incorrect at small $K$ where the drag component is linearly proportional to the incoming velocity with a phase difference of ${\rm \pi} /4$. A general form of the Morison equation is proposed by considering both viscous and form drag components and demonstrated to be superior to the conventional equation for $K < {\sim }2.0$.

We investigate turbulent flow over streamwise-aligned riblets (grooves) of various shapes and sizes. Small riblets with spacings of typically less than $20$ viscous units are known to reduce skin-friction drag compared to a smooth wall, but larger riblets allow inertial-flow mechanisms to appear and cause drag reduction to break down. One of these mechanisms is a Kelvin–Helmholtz instability that García-Mayoral & Jiménez (J. Fluid Mech., vol. 678, 2011, pp. 317–347) identified in turbulent flow over blade riblets. In order to evaluate its dependence on riblet shape and thus gain a broader understanding of the underlying physics, we generate an extensive data set comprising 21 cases using direct numerical simulations of fully developed minimal-span channel flow. The data set contains six riblet shapes of varying sizes between maximum drag reduction and significant drag increase. Comparing the flow fields over riblets to that over a smooth wall, we find that in this data set only large sharp-triangular and blade riblets have a drag penalty associated with the Kelvin–Helmholtz instability and that the mechanism appears to be absent for blunt-triangular and trapezoidal riblets of any size. We therefore investigate two indicators for the occurrence of Kelvin–Helmholtz rollers in turbulent flow over riblets. First, we confirm for all six riblet shapes that the groove cross-sectional area in viscous units serves as a proxy for the wall-normal permeability that is necessary for the development of Kelvin–Helmholtz rollers. Additionally, we find that the occurrence of the instability correlates with a high momentum absorption at the riblet tips. The momentum absorption can be qualitatively predicted using Stokes flow.

To elucidate the key factor for the quantitative prediction of the shear thickening in suspensions in viscoelastic fluids, direct numerical simulations of many-particle suspensions in a multi-mode Oldroyd-B fluid are performed using the smoothed profile method. Suspension flow under simple shear flow is solved under periodic boundary conditions by using Lees–Edwards boundary conditions for particle dynamics and a time-dependent oblique coordinate system that evolves with mean shear flow for fluid dynamics. Semidilute many-particle suspensions up to a particle volume fraction of 0.1 are investigated. The presented numerical results regarding the bulk rheological properties of the shear-thickening behaviour agree quantitatively with recent experimental results of semidilute suspensions in a Boger fluid. The presented result clarifies that an accurate estimation of the first normal stress difference of the matrix in the shear-rate range where the shear thickening starts to occur is crucial for the quantitative prediction of the suspension shear thickening in a Boger fluid matrix at around the Weissenberg number ${Wi}=1$ by an Oldroyd-B model. Additionally, the effect of suspension microstructures on the suspension viscosity is examined. The paper concludes with a discussion on how the flow pattern and the elastic stress development change with the volume fraction and Weissenberg number.

It has been reported that the follower in a tandem configuration with no wall (0W) reduces the time-averaged input power by utilizing the vortex interception mode (Zhu et al., Phys. Rev. Lett., vol. 113, 2014, p. 238105). In the present study, a numerical simulation is conducted with two self-propelled flexible fins in the tandem configuration near a single wall (1W) and two parallel walls (2W). Contrary to the vortex interception for 0W, the follower employs spontaneously a mixed mode (i.e. a combination of the vortex interception mode and the slalom mode) for 1W and the slalom mode for 2W. Although the lateral motion of the follower for 0W, 1W and 2W is synchronized with the induced lateral flow generated by the leader, the time-averaged input power of the follower for 1W and 2W is reduced significantly due to the enhanced lateral flow by the vortex–vortex interaction near the wall. The jet-like flow opposite to the moving direction continuously hinders the movement of the follower for 0W, whereas the follower for 1W and 2W utilizes the negative horizontal flow when passing between the main vortex and the induced vortex near the wall, leading to a decrease of the thrust force acting on the follower allowing the follower to keep pace with the leader. The global efficiency of the schooling fins is optimized with a small heaving amplitude of the follower and a critical value of phase difference between the leader and follower when the values of the wall proximity and bending rigidity are moderate.

The paper is devoted to the study of the periodic rectilinear motion of an elliptic cylinder at an arbitrary angle of attack in a viscous incompressible fluid according to an arbitrary multi-harmonic law. The study develops the classical method of inner and outer asymptotic expansions of a solution in a small parameter, which is taken as the ratio of the thickness of the Stokes boundary layer to the semi-major axis of the elliptic cylinder. Another parameter of the problem, which is the Reynolds number calculated based on the thickness of the boundary layer, is considered to be of the order of unity. The selected range of values of parameters describes the low-amplitude high-frequency oscillations of the cylinder in a fluid. The asymptotic solution of the problem built in the study allows us to determine the first three terms of the expansion of the hydrodynamic force acting on the cylinder. For the detailed analysis of the obtained solution, several cases of the periodic motion of cylinders with different axis ratios for different oscillation parameters according to harmonic and multi-harmonic laws were considered. In these cases, the asymptotic dependencies of the hydrodynamic forces were compared with the data of direct numerical simulations and with known results of previous asymptotic and experimental research. The results of the analysis allow us to establish the limits of applicability of the asymptotic theory and to determine the structural changes of the hydrodynamic forces when transition to the zone of moderate oscillation amplitudes occurs.

Solidifying ternary systems can exhibit complex natural convection phenomena, particularly due to the presence of two porous zones (cotectic and primary mush), and the rejection of two differently dense solutes. The primary objectives of this study are to investigate the following: (i) the natural convection patterns in various compositional regimes of a typical ternary system, and (ii) the role of the combined existence of the microstructure (facets and dendrites) in the porous zone on natural convection, with a motivation to enhance the current understanding of the microstructure–convection relationships. A ternary mixture is chosen such that different compositions of the three primary solidifying components lead to the formation of distinct ice, dendritic and faceted solid structures that cover the complete span of microstructure–convection relationships. The observations of flow in different compositional regimes show convection occurring in the form of plumes, random mixing and double-diffusive layering, as well as combinations of these, which are governed by the type of coexisting microstructures. The study reveals the occurrence of Rayleigh–Taylor instability with varying amounts of the heavier component. The bulk liquid composition showed a tendency to cross the cotectic line, and thus also change the nature of primary solidifying structure from faceted to dendritic in cases where facets and dendrites were present in cotectic mush, and facets in primary mush. These insights are believed to elucidate the complex mechanisms of ternary solidification, as well as provide important real-time data for direct numerical simulations.

The three-dimensional turbulence structure of the canonical daytime atmospheric boundary layer (ABL) reflects the balance between shear and buoyancy turbulence production characterized by $-z_i/L>0$, where $z_i$ is the boundary layer height and $L<0$ is the Obukhov length. In the shear-driven neutral ABL ($-z_i/L=0$) the surface layer is characterized by coherent low-speed streaks, while the ‘moderately convective’ state ($-z_i/L\sim 1\text {--}10$) contains streamwise-elongated sheet-like updraughts. Using large-eddy simulation, we analyse the transition in ABL turbulence structure in response to systematic increases in surface heat flux and $-z_i/L$ from neutral to moderately convective. We discover a sudden change in turbulence structure at the ‘critical’ state $-z_i/L=0.433$ with nearly three-fold increase in the streamwise coherence length of ‘streaks’ at the upper surface layer and four-fold increase in the streamwise scale of updraughts in the mixed layer – as well as the initiation of spatial correlations between the two that contribute to the local generation of thermal updraughts as surface heating increases. At subcritical stability states, buoyancy impacts only the vertical thickness of the streaks. ABL receptivity to buoyancy changes when supercritical and updraughts now grow in the vertical with increasing surface heat flux; the previously amalgamated updraughts extend further in the streamwise direction. The post-critical regime is highlighted by a ‘maximum coherence state’ at $-z_i/L=1.08$ and maximally coherent ‘large-scale rolls’. At increasingly unstable states ($-z_i/L>1.08$), roll coherence decreases and the vertical scales asymptote to the canonical moderately convective ABL.

The instability of Hagen–Poiseuille flow in a rough pipe is considered and it is shown that for arbitrarily small roughness amplitudes the flow is unstable for sufficiently large values of the Reynolds number. Various models of wall roughness are considered and, if $\epsilon$ is a typical amplitude of the roughness, it is shown that the flow is unstable when the Reynolds number $R> C {\epsilon ^{-({3}/{2})} \vert {\log \epsilon }\vert ^{-({3}/{4})}}$ where $C$ is a constant which depends on the roughness shape and is typically in the range 10–40. The roughness is assumed to vary on the same length scale as the pipe radius. In the limit of short scale roughness varying most quickly in the streamwise direction, a quite general condition for instability, $R_b > {3.16 [{b}/{h}]^{3/4}} /{(\log [{b}/{h}])^{3/8}}$, is found in terms of just the Reynolds number $R_b$ based on the friction velocity, the streamwise length scale $b$ and $h$, the height of the roughness. The instability mechanism described is closely linked to vortex–wave interaction theory and applies to both two- and three-dimensional roughness shapes and takes the form of a roll-streak-wave flow. The interaction sustaining the instability occurs in a viscous boundary layer at the pipe wall but the roll-streak flow persists throughout the pipe. The most dangerous roughness shapes are found and generic results are also given for when the roughness length scale is small compared to the pipe radius.

Sucrose is among the main products of photosynthesis that are deemed necessary for plant growth and survival. It is produced in the mesophyll cells of leaves and translocated to different parts of the plant through the phloem. Progress in understanding this transport process remains fraught with experimental difficulties, thereby prompting interest in theoretical approaches and laboratory studies. The Münch pressure and mass flow model is one of the accepted hypotheses describing the physics of sucrose transport in the phloem. It is based on osmosis creating an energy potential difference between the source and the sink. The flow responding to this energy potential is assumed laminar and described by the Hagen–Poiseuille equation. This study revisits such osmotically driven flows in tubes with membrane walls by including the effects of Taylor dispersion on mass transport. This effect has been overlooked in phloem flow studies. Taylor dispersion can increase the effective transport of solutes by increasing the apparent diffusion coefficient. It is shown that, in addition to the conventional diffusive correction derived for impermeable tube walls, a new adjustment to the mean advective terms arises because of osmotic effects. Because the molecular Schmidt number is very large for sucrose in water, the sucrose front speed and travel times have a direct dependence on the Péclet number for different ranges of the Münch number. This study establishes upper limits on expected Taylor dispersion enhancement of sucrose transport.

We present a fully microscopic study of the interfacial thermodynamics of nanodrops and nanobubbles, adsorbed on flat substrates with first-order wetting. We show that both nanodrops and nanobubbles are thermodynamically accessible in regions, demarcated by the spinodals of planar wetting films, with nanobubbles occupying a relatively bigger portion of the phase space. While nanodrops can be described as near-spherical caps of Laplace radius, the radius of nanobubbles is very different from the Laplace value. Additionally, nanobubbles are accompanied by a thin gas film adsorbed on the substrate. By computing the interface binding potential, we relate the sphericity of nanodrops to the thin–thick liquid film coexistence (prewetting transition), whereas nanobubble shapes are determined only by the decay of the fluid–substrate forces.

Conservation laws that relate the local time-rate-of-change of the spatial integral of a density function to the divergence of its flux through the boundaries of the integration domain provide integral constraints on the spatio-temporal development of a field. Here we show that a new type of conserved quantity exists that does not require integration over a particular domain but which holds locally at any point in the field. This is derived for the pseudo-energy density of non-divergent Rossby waves where local invariance is obtained for (i) a single plane wave, and (ii) waves produced by an impulsive point source of vorticity. The definition of pseudo-energy used here consists of a conventional kinetic part, as well as an unconventional pseudo-potential part, proposed by Buchwald (Proc. R. Soc. Lond. A, vol. 328, issue 1572, 1972, pp. 37–48). The anisotropic nature of the energy flux that appears in response to the point source further clarifies the role of the beta plane in the observed western intensification of ocean currents.

Internal solitary waves in the coastal ocean propagate in a complex environment, with variations in the background currents and stratification and with topography along both the bottom (e.g. sills) and sides (e.g. headlands). We present direct numerical simulations of internal solitary wave propagation past an isolated sidewall constriction on laboratory scales. We find that the wave-induced currents generate separation regions which develop into vortices above and below the wave-deformed pycnocline. These vortices yield horizontal tracer exchange between the near-wall region and channel interior and vertical transport sufficient to lift pycnocline fluid near to the surface. Quantitatively, the height of the vertical transport was found to be proportional to the square of the vertical vorticity. Increases in the wave amplitude and the aspect ratio of the constriction (height to width) led to stronger vortices, greater lateral and vertical transport and enhanced density overturning. We compare and contrast these findings with the literature on isolated bottom topography, focusing on the inherently three-dimensional nature (vorticity aligned perpendicular to isopycnals) of the instability in the sidewall case as opposed to typical instabilities caused by internal solitary waves.

The influence of weak low-frequency actuation on the three-dimensional turbulent wake of a cantilevered triangular prism of aspect ratio 4 protruding a thin turbulent boundary layer is investigated experimentally at a Reynolds number of 12 000. Results are reported for surface pressure measurements on the leeward face and wake of the obstacle and particle image velocimetry for selected planes in the wake. Zero-net-mass-flux actuation (sinusoidally modulated synthetic jets) is used to excite the flow through two slits spanning the obstacle height along the edges of the leeward face. Vortex shedding lock-on is shown to occur over intervals corresponding to subharmonics of the actuation frequency. The synchronization mechanism is identified, where weak perturbations due to actuation at critical stages of the Kármán vortex formation trigger shedding. A phenomenological model is presented, linking the concepts of vortex formation time and circulation transport, to describe lock-on phenomena for one-sided and symmetric two-sided actuation. The model further describes interactions with the synthetic jet leading to the splitting of shed vortices observed in earlier studies. Similarities to results observed for other geometric and actuation configurations suggest a broader relevance of the proposed model and highlight differences between weak and strong forcing.

We present a derivation that begins with the Navier–Stokes equation and ends with a prediction of multiple statistically stable states identical to those observed in a spanwise rotating plane Couette flow. This derivation is able to explain the presence of multiple states in fully developed turbulence and the selection of one state over another by differently sized computational domains and different initial conditions. According to the present derivation, two and only two statistically stable states are possible in an infinitely large plane Couette flow with spanwise rotation, and that multiple states are not possible at very slow or very rapid rotation speeds. We also show the existence of limit-cycle-like behaviours near statistically stable states.

This study investigates deterministic wave forecasting from the perspective of the Zakharov equation. Forecasts based on linear dispersion, weakly nonlinear amplitude dispersion and the Zakharov equation are compared for reference ocean surfaces generated from the Joint North Sea Wave Observation Project and Pierson–Moskowitz spectra. This approach allows for the role of nonlinearity to be isolated, and demonstrates the success of simple frequency corrections in forecasting wave fields up to moderate steepness. The role of second-order bound waves is investigated by means of analytical formulae, and their impact on forecast accuracy is illustrated for a range of forecast times.

The experimental and numerical investigation presented by Bertsch et al. (Phys. Rev. Fluids, vol. 5, 2020a, p. 054202) describes the self-sustained oscillations induced by the interaction of two impinging jets in microfluidic devices. While the oscillatory regime induced by interacting jets has been studied in detail, the physical mechanism behind these oscillations remains still undetermined. The present paper focuses on two-dimensional oscillators subjected to a fully developed inlet flow, as in Bertsch et al. (Phys. Rev. Fluids, vol. 5, 2020a, p. 054202) and in contradistinction with Pawlowski et al. (J. Fluid Mech., vol. 551, 2006, pp. 117–139), who focused on plug inlet flow. The linear global stability analysis performed confirms the existence of an oscillating global mode, whose spatial structure qualitatively coincides with that computed numerically by Bertsch et al. (Phys. Rev. Fluids, vol. 5, 2020a, p. 054202), suggesting that the physical mechanism from which the oscillations would originate is predominantly two-dimensional. The interaction of the oscillating mode with a steady symmetry-breaking mode is examined making use of the weakly nonlinear theory, which shows how the system exhibits hysteresis in a certain range of aspect ratios. Lastly, sensitivity analysis is exploited to identify the wavemaker associated with the global modes, whose examination allows us to spot the core of the symmetry-breaking instability at the stagnation point and to identify the Kelvin–Helmholtz instability, located in the jets interaction region, as the main candidate for the origin of the oscillations observed in fluidic devices.