Recent experiments and simulations have sparked growing interest in the study of Rayleigh–Bénard convection in very slender cells. One pivotal inquiry arising from this interest is the elucidation of the flow structure within these very slender cells. Here we employ tomographic particle image velocimetry, for the first time, to capture experimentally the full-field three-dimensional and three-component velocity field in a very slender cylindrical cell with aspect ratio $\Gamma =1/10$. The experiments cover a Rayleigh number range $5.0 \times 10^8 \leqslant Ra \leqslant 5.0 \times 10^9$ and Prandtl number 5.7. Our experiments reveal that the flow structure in the $\Gamma =1/10$ cell is neither in the multiple-roll form nor in the simple helical form; instead, the ascending and descending flows can intersect and cross each other, resulting in the crossing events. These crossing events separate the flow into segments; within each segment, the ascending and descending flows ascend or descend side by side vertically or in the twisting manner, and the twisting is not unidirectional, while the segments near the boundary can also be in the form of a donut like structure. By applying the mode decomposition analyses to the measured three-dimensional velocity fields, we identified the crossing events as well as the twisting events for each instantaneous flow field. Statistical analysis of the modes reveals that as $Ra$ increases, the average length of the segments becomes smaller, and the average number of segments increases from 2.5 to 3.9 in the $Ra$ range of our experiments.

The interaction between a turbulent flow and a porous boundary is analysed with focus on the sensitivity of the roughness function, $\Delta U^+$, to the upscaled coefficients characterizing the wall. The study is aimed at (i) demonstrating that imposing effective velocity boundary conditions at a virtual plane boundary, next to the physical one, can efficiently simplify the direct numerical simulations (DNS); and (ii) pursuing correlations to estimate $\Delta U^+$ a priori, once the upscaled coefficients are calculated. The homogenization approach employed incorporates near-interface advection via an Oseen-like linearization, and the macroscopic coefficients thus depend on both the microstructural details of the wall and a slip-velocity-based Reynolds number, $Re_{slip}$. A set of homogenization-simplified DNS is run to study the channel flow over transversely isotropic porous beds, testing values of the grains’ pitch within $0\lt \ell ^+\lt 40$. Reduction of the skin-friction drag is attainable exclusively over streamwise-aligned inclusions for $\ell ^+$ values up to $20{-}30$. The drag increase over spanwise-aligned inclusions (or streamwise-aligned ones at large $\ell ^+$) is accompanied by enhanced turbulence levels, including intensified sweep and ejection events. The root-mean-square of the transpiration velocity fluctuations at the virtual plane, $\tilde V_{rms}$, is the key control parameter of $\Delta U^+$; our analysis shows that, provided $\tilde V_{rms} \lesssim 0.25$, then $\tilde V_{rms}$ is strongly correlated to a single macroscopic quantity, $\Psi$, which comprises the Navier-slip and interface/intrinsic permeability coefficients. Fitting relationships for $\Delta U^+$ are proposed, and their applicability is confirmed against reference results for the turbulent flow over impermeable walls roughened with three-dimensional protrusions or different geometries of riblets.

A new temporal vortex tracking algorithm allows the first long-term temporal observation of the lives of the intense vorticity structures (IVS). The algorithm is applied to direct numerical simulations of statistically stationary isotropic turbulence, with Taylor-based Reynolds numbers in the range $54 \leqslant Re_{\lambda } \leqslant 239$. In the highest Reynolds number case, the continuous time tracking of millions of ‘worms’ is achieved for more than seven integral time scales and close to 200 Kolmogorov time scales. Within an integral scale volume, more than 66 structures exist, and approximately 20 new structures are created per Kolmogorov time. More than $80\, \%$ of the structures live a solitary ‘life’ without any visible interaction with the other structures, while approximately $15\, \%$ break into new structures. Less than $2\, \%$ of the structures merge with others to form new vortices. A ‘population model’ is developed to estimate the numbers of existing vortices for very long simulated times, and it is observed that the birth rate density of these structures slowly increases with the Reynolds number. The survival functions of the IVS lives exhibit an exponential distribution, with some structures living for more than $35$ Kolmogorov time scales (more than four integral time scales). The mean lifetime of the IVS scales with the mean turnover time scale of the worms, defined by their radii and tangential velocity, attaining $\approx 6.5$ turnover time scales at high Reynolds numbers.

Liquid metal flows are important for many industrial processes, including liquid metal batteries (LMBs), whose efficiency and lifetime can be affected by fluid mixing. We experimentally investigate flows driven by electrical currents in an LMB model. In our cylindrical apparatus, we observe a poloidal flow that descends near the centreline for strong currents, and a poloidal flow that rises near the centreline for weak currents. The first case is consistent with electrovortex flow, which is an interaction between current and its own magnetic field, whereas the second case is consistent with an interaction between current and the external field, which drives Ekman pumping. Notably, we also observe an intermediate case where the two behaviours appear to compete. Comparing results with Frick et al. (2022 J. Fluid Mech. 949, A20), we test prior estimates of the scaling of flow speed with current to predict the observed reversal. Based on these data, we propose two different ways to apply the Davidson et al. (1999 J. Fluid Mech. 245, 669–699) poloidal suppression theory that explain both experimental results simultaneously: either taking the wire radius into account to scale the Lorentz force, or taking viscous dissipation into account to scale the swirl velocity, following Herreman et al. (2021 J. Fluid Mech. 915, A17).

Self-similar fractal tree models are numerically investigated to elucidate the drag coefficient, non-equilibrium dissipation behaviour and various turbulence statistics of fractal trees. For the simulation, a technique based on the lattice Boltzmann method with a cumulant collision term is used. Self-similar fractal tree models for aerodynamic computations are constructed using parametric L-system rules. Computations across a range of tree-height-based Reynolds numbers $Re_H$, from 2500 to 120 000, are performed using multiple tree models. As per the findings, the drag coefficients ($C_D$) of these models match closely with those of the previous literature at high Reynolds numbers ($Re_H \geqslant 60\,000$). A normalization process that collapses the turbulence intensity across various tree models is formulated. For a single tree model, a consistent centreline turbulence intensity trend is maintained in the wake region beyond a Reynolds number of 60 000. The global and local isotropy analysis of the turbulence generated by fractal trees indicates that, at high Reynolds numbers ($Re_H \geqslant 60\,000$), the distant wake can be considered nearly locally isotropic. The numerical results confirm the non-equilibrium dissipation behaviour demonstrated in previous studies involving space-filling fractal square grids. The non-dimensional dissipation rate $C_\epsilon$ does not remain constant; instead, it becomes approximately inversely proportional to the local Taylor-microscale-based Reynolds number, $C_\epsilon \propto 1/Re_\lambda$. We find significant one-point inhomogeneity, production and transverse transport of turbulent kinetic energy within the non-equilibrium dissipation near wake region.

We present a linear analysis of a minimal model of moist convection under a variety of atmospheric conditions. The stationary solutions that we analyse include both fully saturated and partially unsaturated atmospheres in both unconditionally and conditionally unstable cases. We find that all of the solutions we consider are linearly unstable via exchange of stability when sufficiently driven. The critical Rayleigh numbers vary by over an order of magnitude between unconditionally unstable and conditionally unstable atmospheres. The unsaturated atmospheres are notable for the presence of linear gravity wave-like oscillations even in unstable conditions. We study their eigenfunction structure and find that the buoyancy and moisture perturbations are anticorrelated in $z$, such that regions of negative buoyancy have positive moisture content. We suggest that these features in unsaturated atmospheres may explain the phenomenon of gravity wave shedding by moist convective plumes.

The bi-stable dynamics of a one-degree-of-freedom disk pendulum swept by a flow and allowed to rotate in the cross-flow direction is investigated experimentally. For increasing flow velocity, a subcritical bifurcation is observed from a Pendulum state, characterised by an increasing time-averaged pendulum angle with large amplitude fluctuations, to a rotating state with a non-zero mean rotation velocity at a critical free stream velocity $U_{P2W}$. The rotating state, referred to as Windmill state, presents a strong hysteresis: once initiated, it is sustained down to velocities $U_{W2P}\lt U_{P2W}$ before bifurcating towards the Pendulum state. A thorough experimental characterisation of the dynamical features of each state is reported, with a particular focus on the influence of the static yaw angle of the disk $\beta _0$ and the free stream velocity. In the Pendulum state, the system behaves differently depending on whether $\beta _0$ lies below or above the stall angle of the disk, with more regular dynamics below. We demonstrate that the bifurcation between the Pendulum state and the Windmill state is triggered by aerodynamic fluctuations, while the bifurcation between the Windmill state and the Pendulum state is deterministic. A stochastic model faithfully reproduces the dynamical features of both states, as well as the characteristics of the bifurcations.

Microorganisms, such as spermatozoa, exhibit rich behaviours when in close proximity to each other. However, their locomotion is not fully understood when coupled mechanically and hydrodynamically. In this study, we develop hydrodynamic models to investigate the locomotion of paired spermatozoa, predicting the fine structure of their swimming. Experimentally, sperm pairs are observed to transition between different modes of flagellar synchronisation: in-phase, anti-phase and lagged synchronisation. Using our models, we assess their swimming performances in these synchronisation modes in terms of average swimming speed, average power consumption, and swimming efficiency. The swimming performances of paired spermatozoa are shown to depend on their flagellar phase lag, flagellar waveforms, and the mechanical coupling between their heads.

In this study, we conducted interface-capturing high-resolution simulations of a bubbly upflow in a vertical channel to investigate the bubble distribution and its interaction with surrounding turbulence, focusing on the effects of the density ratio. A bulk Reynolds number $Re_b=2300$ was used for all simulations. The influence of density ratio on vortex structures and turbulence statistics differed between the near-wall and core regions of the channel. Adding 5.43 $\%$ gas caused an increase in wall friction. By applying a generalised FIK identity to analyse wall friction, it was determined that the drag rise in the bubbly channel was mostly due to the near-wall region. Visualisation of the bubble and vortex structures showed that small bubbles near the wall induced larger magnitude of Reynolds shear stress and increased wall friction. Bubble behaviour near the wall region was similar for density ratios above 30, leading to wall friction saturation. In the core region, large deformable bubbles created wake vortices due to slip velocity between liquid and gas phases. Wake vortices help large bubbles absorb smaller bubbles and maintain their sizes. As the density ratio increased, the slip velocity increased owing to greater difference in the gravitational acceleration between liquid and gas phases, resulting in corresponding increase in wake intensity and velocity fluctuations. However, quadrant analysis showed that Q1 and Q3 events increased together with Q2 and Q4 events in the core region, cancelling out any net effect of wake vortices on Reynolds shear stress or wall friction.

Highly resolved simulations reveal the fundamental influence of a carrier fluid’s flow dynamics on triboelectric powder charging. We found that particles transported through a square-shaped duct charge faster than in a channel flow caused by secondary flows that led to more severe particle–wall collisions. Specifically, particles with a Stokes number of 4.69 achieve 85 % of their equilibrium charge approximately 1.5 times faster in duct flow than in channel flow. Also, charge distribution is more uniform in a duct cross-section compared with a channel cross-section. In channel flow, particles are trapped near the walls and collide frequently due to limited movement in the wall-normal direction, causing localized charge buildup. In contrast, duct flow promotes better mixing through secondary flows, reducing repeating collisions and providing uniform charge distribution across the cross-section. Upon charging, electrostatic forces significantly reshape particle behaviour and distribution. Once the powder achieves half of its equilibrium charge, particles increasingly accumulate at the wall, leading to a reduced concentration in the central region. These changes in particle distribution have a noticeable impact on the surrounding fluid phase and alter the overall flow dynamics. These findings open the possibility for a new measure to control powder charging by imposing a specific pattern.

Real-time wave forecasting (RTWF) consists in predicting ocean wave motion or forces, from seconds to minutes in advance, using real-time measurements. For the successful development of RTWF, understanding wave predictability is essential. Usually, a deterministic ‘predictable zone’ (DPZ) is geometrically constructed from the wave group velocities and directions present in the spectrum. DPZs have little experimental evidence, and suffer ambiguities regarding the choice of cutoff frequencies and directions – since actual ocean waves are not band-limited. The present study addresses those shortcomings, by defining probabilistic predictable zones (PPZs) with respect to chosen uncertainty thresholds, using a rigorous statistical framework restricted to near-Gaussian sea states (precisely those where RTWF would be employed). PPZs are examined in idealised spectra and in a stereo dataset of a real wave field. It is shown that the PPZ geometry is quantitatively related to the sea state characteristics, through three physical parameters: two limiting group velocities (similar to the deterministic theory), and a directional spreading effect, which also limits the PPZ extent. While the lower group velocity depends on the chosen uncertainty threshold, the upper group velocity is better approximated by that of the spectrum peak frequency, which is a novel finding. The empirical data support the validity of the present PPZ theory. In contrast, both theoretical and empirical results contradict the fan-shaped predictable zones, constructed in the three-dimensional deterministic theory, thus highlighting the importance of considering stochasticity to understand the predictability of actual ocean waves.

We report laboratory experiments of long-crested irregular water surface waves propagating over a shoal, with attention to the region over the down-slope behind the shoal. We measure the surface elevation field, the horizontal velocity field in the water, and the resulting forces on a horizontal submerged cylinder placed over the down-slope of the shoal. In addition, we calculate the horizontal acceleration field. From this, we find that the presence of the shoal can modify the wave field such that the resulting forces on the submerged cylinder can be enhanced with thicker extreme tails and increased values of skewness and kurtosis depending on the location of the cylinder. The spatial dependence of the statistics of forces is different from the spatial dependence of the statistics of horizontal velocity, horizontal acceleration and surface elevation.

A pressure-gradient-induced laminar separation bubble (LSB) was examined using wind-tunnel experiments, direct numerical simulations (DNS) and linear local/global stability analysis. The LSB was experimentally generated on a flat plate using the favourable-to-adverse pressure gradient imposed by an inverted modified NACA $64_3-618$ airfoil. Direct numerical simulation was performed using boundary conditions extracted from a steady precursor simulation of the entire flow field. Despite good agreement in the upstream boundary layer with the experiment, DNS exhibited an approximately 25 % longer mean separation bubble, attributed to an earlier onset of transition due to the free-stream turbulence (FST) in the experiment. Introducing a very low level of isotropic FST in the DNS, similar to that measured in the experiment, caused earlier transition, decreased the mean bubble length and led to a remarkably good agreement between the DNS and experiments. Differences were observed for the dominant frequencies in the experiment and DNS, but both were within the band of most amplified frequencies predicted by LST. Proper orthogonal decomposition confirmed that dominant coherent structures from DNS and experiments are related to the inviscid Kelvin–Helmholtz instability and have similar characteristics despite slight differences in frequency. Local and global stability and dynamic mode decomposition analysis corroborated the convective nature of the dominant two-dimensional (2-D) instability and showed that the LSB is globally unstable to a range of 3-D wavenumbers, in agreement with 3-D structures observed in experiments. Results confirmed the strong impact of very low FST levels on the LSB and indicate a close agreement of the time-averaged and instability characteristics between the experiments and DNS.

The incompressible Navier–Stokes equations in spherical coordinates are solved using a pseudo-spectral method to simulate the problem of spherical Couette flow. The flow is investigated for a narrow-gap ratio with only the inner sphere rotating. We find that the flow is sensitive to the initial conditions and have used various initial conditions to obtain different branches of the bifurcation curve of the flow. We have identified three different branches dominated respectively by axisymmetric flow, travelling wave instability and equatorial instability. The axisymmetric branch shows unsteadiness at large Reynolds numbers. The travelling wave instability branch shows spiral instability and is prominent the near poles. The travelling wave instability branch further exhibits a reversal in the propagation direction of the spiral instability as the Reynolds number is increased. This branch also exhibits a multi-mode equatorial instability at larger Reynolds numbers. The equatorial instability branch exhibits twin jet streams on either side of the equator, which become unstable at larger Reynolds numbers. The flow topology on the three branches is also investigated in their phase space and found to exhibit chaotic behaviour at large Reynolds numbers on the travelling wave instability branch.

Combustion instability analysis in annular systems often relies on reduced-order models that represent the complexity of combustion dynamics in a framework in which the flame is represented by a ‘flame describing function’ (FDF), portraying its heat release rate response to acoustic disturbances. However, in most cases, FDFs are only available for a limited range of disturbance amplitudes, complicating the description of the saturation process at high oscillation levels leading to the establishment of a limit cycle. This article shows that this difficulty may be overcome using a novel experimental scheme, relying on injector staging and in which the oscillation amplitude at limit cycle can be controlled, enabling us to measure FDFs from simultaneous pressure and heat release rate recordings. These data are then exploited to replace the standard modelling, in which the heat release rate is expressed as a third-order polynomial of pressure fluctuations, by a function of the modulation amplitude, allowing an easier adaptation to experimental data. The FDF is then used in a dynamical framework to analyse a set of staging configurations in an annular combustor, where two families of injectors are mixed and form different patterns. The limit-cycle amplitudes and the coupling modes observed experimentally are suitably retrieved. Finally, an expression for the growth rate is derived from the slow-flow variable equations defining the modal amplitudes and phase functions, which is shown to exactly agree with that obtained previously by using acoustic energy principles, providing a theoretical link between growth rates and limit-cycle amplitudes.

A generalized reciprocal theorem is used to relate the force and torque induced on a particle in an inertia-less fluid with small variation in viscosity to integrals involving Stokes flow fields and the spatial dependence of viscosity. These resistivity expressions are analytically evaluated using spheroidal harmonics and then used to obtain the mobility of the spheroid during sedimentation, and in linear flows, of a fluid with linear viscosity stratification. The coupling between the rotational and translational motion induced by stratification rotates the spheroid’s centerline, creating a variety of rotational and translational dynamics dependent upon the particle’s aspect ratio, $\kappa$, and the component of the stratification unit vector in the gravity direction, $d_g$. Spheroids with $0.55\lessapprox \kappa \lessapprox 2.0$ exhibit the largest variety of settling behaviors. Interestingly, this range covers most microplastics and typical microorganisms. One of the modes include a stable orientation dependent only on $\kappa$ and $d_g$, but independent of initial orientation, thus allowing for the potential control of settling angles and sedimentation rates. In a simple shear flow, cross-streamline migration occurs due to the stratification-induced force generated on the particle. Similarly, a particle no longer stays at the stagnation point of a uniaxial extensional flow. While fully analytical results are obtained for spheroids, numerical simulations provide a source of validation. These simulations also provide additional insights into the stratification-induced force- and torque-producing mechanisms through the stratification-induced stress, which is not accessed in the reciprocal theorem-based analytical calculations.

We investigate the stability of a compressible boundary layer over an impedance wall for both constant impedances and a frequency-dependent porous wall model. For an exponential mean flow profile, the solution of the Pridmore-Brown equation, i.e. the linearised Euler equations for compressible shear flows, is expressed exactly in confluent Heun functions and, with the boundary condition of acoustic wall impedance, reduced to a single algebraic eigenvalue equation. This, in turn, is solved asymptotically and numerically and provides the complete inviscid eigenvalue spectrum without spurious modes. The key finding is that impedance walls not only have a desirable stabilising effect on inviscid disturbances, but also induce new instabilities. The type of the destabilised mode and therefore also the direction of propagation of the modes with maximum growth rate as well as the destabilised wavenumbers depend significantly on the porous wall properties, in particular on the porous wall layer thickness. For small porous layer thicknesses, the impedance-induced instability is observed as a second mode instability, where we find above a critical porosity growth rates exceeding those present in the rigid-wall case.

Turbulence beneath a free surface leaves characteristic long-lived signatures on the surface, such as upwelling ‘boils’, near-circular ‘dimples’ and elongated ‘scars’, easily identifiable by eye, e.g. in riverine flows. In this paper, we analyse data from direct numerical simulations to explore the connection between these surface signatures and the underlying vortical structures. We investigate dimples, known to be imprints of surface-attached vortices, and scars, which have yet to be extensively studied, by analysing the conditional probabilities that a point beneath a signature is within a vortex core as well as the inclination angles of sub-signature vorticity. The analysis shows that the probability of vortex presence beneath a dimple decreases from the surface down through the viscous and blockage layers. This vertical variation in probability is approximately a Gaussian function of depth and depends on the dimple’s size and the bulk turbulence properties. Conversely, the probability of finding a vortex beneath a scar increases sharply from the surface to a peak at the edge of the viscous layer, regardless of scar size. The probability distributions of the angle between the vorticity vector and the vertical axis also show a clear pattern about vortex orientation: a strong preference for vertical alignment below dimples and an equally strong preference for horizontal alignment below scars. Our findings corroborate previous studies that tie dimples to surface-attached vertical vortices. Moreover, they suggest that scars can be defined as imprints of horizontal vortices that are located approximately a quarter of the Taylor microscale beneath the free surface.

The encapsulation of active particles, such as bacteria or active colloids, inside a droplet gives rise to a non-trivial shape dynamics and droplet displacement. To understand this behaviour, we derive an asymptotic solution for the fluid flow about a deformable droplet containing an active particle, modelled as a Stokes-flow singularity, in the case of small shape distortions. We develop a general solution for any Stokes singularity and apply it to compute the flows and resulting droplet velocity due to common singularity representations of active particles, such as Stokeslets, rotlets and stresslets. The results show that offsetting of the active particle from the centre of the drop breaks symmetry and excites a large number of generally non-axisymmetric shape modes as well as particle and droplet motion. In the case of a swimming stresslet singularity, a run-and-tumble locomotion results in superdiffusive droplet displacement. The effect of interfacial properties is also investigated. Surfactants adsorbed at the droplet interface counteract the internal flow and arrest the droplet motion for all Stokes singularities except the Stokeslet. Our results highlight strategies to steer the flows of active particles and create autonomously navigating containers.

The evolution of two-phase structures, turbulence/dust concentration structures, during an entire sandstorm process, including non-stationary flow, has been originally investigated in this study. Dust concentration structures are observed at different sandstorm stages, which are similar to the turbulence structures. These two-phase structures adhere to self-similarity in the steady stage but fail in the non-stationary stage. However, dust particle exhibits a better capability to follow eddies in flow, but the evolution of dust structures is not analogous to that of turbulence structures, exhibiting distinct trends. Dust particles, initiated from the ground, gradually form cluster structures in the rising stage. Their morphology exhibits a ridge-like evolutionary trend, reaching a peak in the steady stage. In contrast, turbulence structures are most persistent and oblique in the early stage but sequentially diminish in the subsequent steady and declining stages. The significant changes in shear due to sharply varying wind velocity and thermal stability are primarily responsible for these evolution differences.