Applying a variational analysis, a minimum-enstrophy vortex in three-dimensional (3-D) fluids with continuous stratification is found, under the quasi-geostrophic hypothesis. The buoyancy frequency is held constant. This vortex is an ideal limiting state in a flow with an enstrophy decay while energy and generalized angular momentum remain fixed. The variational method used to obtain two-dimensional (2-D) minimum-enstrophy vortices is applied here to 3-D integral quantities. The solution from the first-order variation is expanded on a basis of orthogonal spherical Bessel functions. By computing second-order variations, the solution is found to be a true minimum in enstrophy. This solution is weakly unstable when inserted in a numerical code of the quasi-geostrophic equations. After a stage of linear instability, nonlinear wave interaction leads to the reorganization of this vortex into a tripolar vortex. Further work will relate our solution with maximal entropy 3-D vortices.

Steady streaming is a peculiar flow, induced by the oscillation of an object with respect to the fluid in which it is embedded. While the oscillation amplitude may be extremely small, the nonlinearity due to fluid inertia gives rise to this steady flow. Large and powerful vortices develop around the object. Usually, the streaming is scrutinised in the case of homogeneous objects. Here, Cui et al. (J. Fluid Mech., vol. 979, 2024, A7) derive the streaming around a soft elastic sphere containing a solid core. This elaborate derivation opens a new description of streaming in biological soft environments and robotics.

Thin airfoil dynamic stall at moderate Reynolds numbers is typically linked to the sudden bursting of a small laminar separation bubble close to the leading edge. Given the strong sensitivity of laminar separation bubbles to external disturbances, the onset of dynamic stall on a NACA0009 airfoil section subject to different levels of low-amplitude free stream disturbances is investigated using direct numerical simulations. The flow is practically indistinguishable from clean inflow simulations in the literature for turbulence intensities at the leading edge of ${Tu} = 0.02\,\%$. At slightly higher turbulence intensities of ${Tu} = 0.05\,\%$, the bursting process is found to be considerably less smooth and strong coherent vortex shedding from the laminar separation bubble is observed prior to the formation of the dynamic stall vortex (DSV). This phenomenon is considered in more detail by analysing its appearance in an ensemble of simulations comprising statistically independent realisations of the flow, thus proving its statistical relevance. In order to extract the transient dynamics of the vortex shedding, the classical proper orthogonal decomposition method is generalised to include time in the energy measure and applied to the time-resolved simulation data of incipient dynamic stall. Using this technique, the dominant transient spatiotemporally correlated features are distilled and the wave train of the vortex shedding prior to the emergence of the main DSV is reconstructed from the flow data exhibiting dynamics of large-scale coherent growth and decay within the turbulent boundary layer.

The validity of the Oberbeck–Boussinesq (OB) approximation in Rayleigh–Bénard (RB) convection is studied using the Gray & Giorgini (Intl J. Heat Mass Transfer, vol. 19, 1976, pp. 545–551) criterion that requires that the residuals, i.e. the terms that distinguish the full governing equations from their OB approximations, are kept below a certain small threshold $\hat {\sigma }$. This gives constraints on the temperature and pressure variations of the fluid properties (density, absolute viscosity, specific heat at constant pressure $c_p$, thermal expansion coefficient and thermal conductivity) and on the magnitudes of the pressure work and viscous dissipation terms in the heat equation, which all can be formulated as bounds regarding the maximum temperature difference in the system, $\varDelta$, and the container height, $L$. Thus for any given fluid and $\hat {\sigma }$, one can calculate the OB-validity region (in terms of $\varDelta$ and $L$) and also the maximum achievable Rayleigh number ${{Ra}}_{max,\hat {\sigma }}$, and we did so for fluids water, air, helium and pressurized SF$_6$ at room temperature, and cryogenic helium, for $\hat {\sigma }=5\,\%$, $10\,\%$ and $20\,\%$. For the most popular fluids in high-${{Ra}}$ RB measurements, which are cryogenic helium and pressurized SF$_6$, we have identified the most critical residual, which is associated with the temperature dependence of $c_p$. Our direct numerical simulations (DNS) showed, however, that even when the values of $c_p$ can differ almost twice within the convection cell, this feature alone cannot explain a sudden and strong enhancement in the heat transport in the system, compared with its OB analogue.

We conducted experimental investigations on the effect of stable thermal conditions on rough-wall boundary layers, with a specific focus on their response to abrupt increases in surface roughness. For stably stratified boundary layers, a new analytical relation between the skin-friction coefficient, $C_f$, and the displacement thickness was proposed. Following the sharp roughness change, the overshoot in $C_f$ is slightly enhanced in stably stratified layers when compared with that of neutral boundary layers. Regarding the velocity defect law, we found that the displacement thickness multiplied by $\sqrt{2/C_f}$, performs better than the boundary layer thickness alone when describing the similarity within internal boundary layers for both neutral and stable cases. A non-adjusted region located just beneath the upper edge of the internal boundary layer was observed, with large magnitudes of skewness and kurtosis of streamwise and wall-normal velocity fluctuations for both neutral and stable cases. At a fixed wall-normal location, the greater the thermal stratification, the greater the magnitudes of skewness and kurtosis. Quadrant analysis revealed that the non-adjusted region is characterised by an enhancement/reduction of ejection/sweep events, particularly for stably stratified boundary layers. Spatially, these ejections correspond well with peaks of kurtosis, exhibit stronger intensity and occur more frequently following the abrupt change in surface conditions.

This study presents a new density filter for a pipe-shaped structure and its application to fluid topology optimization. A simple and straight pipe-shaped structure for fluid is preferred for many engineering purposes rather than the perplex manifold structure provided by the topology optimization method. To determine an optimal pipe structure for fluid, we develop a new density filter and apply it to fluid topology optimization. Hence, the original spatially varying design variables of the fluid topology optimization are modified based on the pipe density filter. Subsequently, the filter design variables, including a uniform pipe wall thickness and adjusted cross-section, are used for artificial pseudo-rigid bodies in fluid topology optimization. An additional constraint is imposed to maintain a nearly uniform pipe thickness. Several numerical examples are solved to demonstrate the validity of the present pipe density filter for fluid topology optimization problems minimizing the energy dissipation of the fluid and controlling the particles suspended in the fluid.

Events of extreme intensity in turbulent flows from atmospheric to industrial scales have a strong social and economic impact, and hence there is a need to develop models and indicators which enable their early prediction. Part of the difficulty here stems from the intrinsic sensitivity to initial conditions of turbulent flows. Despite recent progress in understanding and predicting extreme events, the question of how far in advance they can be ideally predicted (without model error and subject only to uncertainty in the initial conditions) remains open. Here we study the predictability limit of extreme dissipation bursts in the two-dimensional Kolmogorov flow by applying information-theoretic measures to massive statistical ensembles with more than $10^7$ direct numerical simulations. We find that extreme events with similar intensity and structure can exhibit disparate predictability due to different causal origins. Specifically, we show that highly predictable extreme events evolve from distinct large-scale circulation patterns. We thus suggest that understanding all the possible routes to the formation of extreme events is necessary to assess their predictability.

A general rethinking of the mathematical foundations of water surface waves from the perspective of the Hilbert transform uncovers shortcomings of the standard multiple-scale approach as well as elucidates the interplay of non-local and dispersive effects. Application of the Hilbert transforms to planar and cylindrical settings allows us to deduce new weakly nonlinear models, including an alternative to Zakharov's equation and an envelope equation for cylindrical waves on deep water, as well as to highlight the crucial differences between these geometries.

The response of airfoils to unsteady disturbances is a classic problem in the aerodynamics field. Many theoretical models have been proposed in the past to predict the unsteady aerodynamic forces of airfoils. However, these theories focused on individual airfoil motions or incoming flow disturbances, while the theoretical models for multiple disturbances still need to be developed. In this study, a theoretical model to predict the aerodynamic force of an oscillating airfoil encountering vertical gust is derived from a linear combination of Theodorsen's and Sears’ theories. Experimental investigations involving a two-dimensional pitching airfoil encountering a sinusoidal vertical gust are carried out to examine the proposed theory. It is found that the theory effectively captures the trends in the unsteady lift of airfoils subjected to dual disturbances. However, it tends to overestimate the lift amplitude. Notably, when a quasi-steady correction is applied to the theory, the prediction accuracy is greatly improved. The theory correction agrees well with experiment at small pitching frequencies, while deviations exist at higher pitching frequencies. The temporal evolution of the flow velocity reveals that the velocity disturbance induced by the coupled disturbance around the airfoil conforms to the linear superposition of the velocities induced by each individual disturbance, consistent with the prediction of the vortex sheet model. As the pitching frequency increases, significant nonlinear effects appear near the trailing edge of the airfoil, which may be one key factor for the disparities between the theoretical predictions and the experimental lift at higher pitching frequencies.

The parameterization of fluxes associated with representing unresolved dynamics in turbulent flows, especially in the atmosphere and ocean (which have a vast range of scales), remains a challenging task. This is especially true for Earth system models including complex biogeochemistry and requiring very long simulations. The problem of representing the dependence of the mean flux of a passive tracer in terms of the mean has a very long history; in this study, we take a somewhat different approach. We use a formalism showing that the mean flux will be a functional of the mean gradients, a formalism that can be used to calculate the structure of the functional which is non-local in both space and time. Two-dimensional turbulent simulations are used to explore the weight of nearby (in space or time) gradients. We also use stochastic velocities and iterated maps to show that the results are similar. The functional formalism provides an understanding of when non-locality needs to be considered and when a local eddy diffusivity can be a reasonably good approximation. Furthermore, the formalism provides guidance for the development of data-driven parameterizations.

Magnetic soap films under the forcing of an inhomogeneous magnetic field are governed by a wide range of interconnected physics. The study of magnetic soap films requires the development of comprehensive models to support experimental observations. In this study, the thin film approximation is applied to the Navier–Stokes equations to derive a model for the film thickness of magnetic soap films that incorporates the effects of interfacial mobility, surfactant transport and magnetite nanoparticle (NP) transport. This derived model consists of a coupled system of equations for the film thickness, interfacial velocity, interfacial surfactant concentration and magnetite NP concentration. Simulations are performed for both soap films and magnetic soap films by solving the system of equations using the Galerkin finite element method, and results are compared with experiments. Simulation results highlight that interfacial flows can dominate the rate of film thinning and that accounting for the dependence of the magnetisation on the local magnetite NP concentration can influence the predicted speed of magnetically driven flows. Furthermore, simulation results demonstrate that the model is able to predict marginal regeneration in qualitative agreement with the experiments for soap films; the model also predicts the same flow pattern as seen in the experiments for magnetic soap films. Overall, this study advances the state of soap film and magnetic soap film modelling and will contribute to acquiring control over the drainage and stability of magnetic soap films in the long term.

Successive drops of coloured ink mixed with surfactant are deposited onto a thin film of water to create marbling patterns in the Japanese art technique of Suminagashi. To understand the physics behind this and other applications where surfactant transports adsorbed passive matter at gas–liquid interfaces, we investigate the Lagrangian trajectories of material particles on the surface of a thin film of a confined viscous liquid under Marangoni-driven spreading by an insoluble surfactant. We study a model problem in which several deposits of exogenous surfactant simultaneously spread on a bounded rectangular surface containing a pre-existing endogenous surfactant. We derive Eulerian and Lagrangian formulations of the equations governing the Marangoni-driven surface flow. Both descriptions show how confinement can induce drift and flow reversal during spreading. The Lagrangian formulation captures trajectories without the need to calculate surfactant concentrations; however, concentrations can still be inferred from the Jacobian of the map from initial to current particle position. We explore a link between thin-film surfactant dynamics and optimal transport theory to find the approximate equilibrium locations of material particles for any given initial condition by solving a Monge–Ampère equation. We find that as the endogenous surfactant concentration $\delta$ vanishes, the equilibrium shapes of deposits using the Monge–Ampère approximation approach polygons with corners curving in a self-similar manner over lengths scaling as $\delta ^{1/2}$. We explore how Suminagashi patterns may be produced by using computationally efficient successive solutions of the Monge–Ampère equation.

The flow-induced vibrations (FIVs) of two identical tandem square cylinders with mass ratio m* = 3.5 at Reynolds number Re = 150 are investigated through two-dimensional direct numerical simulation (DNS) and linear stability analysis over a parameter range of spacing ratio 1.5 ≤ L* ≤ 5 and reduced velocity 3 ≤ Ur ≤ 34. Three kinds of FIV responses, namely vortex-induced vibration (VIV), biased oscillation (BO) and galloping (GA), are identified. The FIVs are then further classified into the branches of initial VIV (IV), resonant VIV (RV and RV′), flutter-induced VIV (FV), desynchronized VIV (DV), VIV developing from GA (GV), transitional state between VIV and GA (TR), BO and GA based on the characteristics of the vibration responses. The transitions among different FIV branches are examined by combining the DNS with linear stability analysis, where the transition boundaries among the VIV, BO and GA branches over the concerned parameters are identified on the branch maps. The transition from IV to RV or RV′ is found to be related to the unstable wake mode, while the FV, transiting from RV or RV′, is induced by the unstable structural factor in the wake-structure mode. The structural instability is considered as the physical origin of GA, whereas the mode competition between unstable wake and structure leads to DV, GV and TR, and thus delays the appearance of GA. The transition from DV to BO with biased equilibrium position, accompanied by the even-order harmonic frequencies, is essentially induced by the symmetry breaking bifurcation.

Direct numerical simulations of spiral Poiseuille flows in a narrow gap geometry are performed with the aim of identifying the mechanisms governing the dynamics of the axial friction coefficient. The investigation has explored a small portion of the Reynolds number–Taylor number phase space ($600 \leq Re \leq 5766$ and $1500 \leq Ta \leq 5000$), for which reference experimental results are available. The study is focused on the mechanism leading to the enhancement of the axial friction coefficient with the Taylor number when the Reynolds number is kept constant. The analysis of the spatial distribution of the Reynolds stress tensor and of the turbulent energy budget has evidenced the key role of the pressure–strain correlation in the energy transfer from the azimuthal to the axial component. The latter eventually determines the increase of the axial friction coefficient through the enhanced radial mixing of axial momentum. Data have also shown that the flow dynamics is heavily dependent on the $Ta/Re$ ratio, and different regimes develop (ranging from laminar to turbulent), each with peculiar behaviours.

Numerical simulations are carried out on the vortex-induced rotations of a freely rotatable rigid square cylinder in a two-dimensional uniform cross-flow. A range of Reynolds numbers between 40 and 150 and density ratios between 0.1 and 10 are considered. Results show eight different characteristic regimes, expanding the classification of Ryu & Iaccarino (J. Fluid Mech., vol. 813, 2017, pp. 482–507). New regimes include the transition and wavy rotation regimes; in the ${\rm \pi}$-limited oscillation regime we observe multipeak subregimes. Moment-generating mechanisms of these regimes and subregimes are further elucidated. A phenomenon related to the influence of density ratio is the tooth-like shape of the ${\rm \pi} /2$-limit oscillation regime observed in the regime map, which is explained as a result of the imbalance relation between the main frequencies of rotation response and the vortex shedding frequency. In addition, existence of multiple regimes and multistable states are discussed, indicating multiple stable attractive structures in phase space.

A wide range of environmental, energy, medical and biological processes rely on dispersive transport through complex media. Yet, because of the stagnant and opaque nature of the microscopic system, the role of disordered flow and structure in the dispersive transport of solutes remains poorly understood. Here, we use a circular porous microfluidic system to investigate the radial dispersion in porous media driven by non-Newtonian fluids with strong advection rate (or at high Péclet number) and low-to-moderate Reynolds numbers. We observe for the first time the presence of diffusion ‘blind zones’ in the microstructure for high solution injection velocities. More specifically, an in-depth analysis uncovers that the circumferential flow frame, coformed by obstacles and vortices especially the ‘twin-vortex’ with same rotation direction, is responsible for the diffusion ‘blind zones’ and transport heterogeneity. The vortices are induced by the coupling of microfluidics and porous structures, and correlated to inertial flow-induced instabilities. The trade-off between diffusion efficiency and quality/completeness with respect to the high Péclet number (or high inlet velocity) serves to enhance our comprehension of intricate fluid dynamics and affords a set of principles to aid a diverse range of practical implementations.

We investigate the influence of vortices remote from the boundary on the near-wall flow dynamics in wall-bounded flows. A vortex ring with precisely controlled local twist is introduced into the outer layer of a channel flow at a moderate Reynolds number. We find that the minimum vorticity flux for triggering the transition to turbulence is significantly reduced from the initial disturbance of an untwisted vortex ring to that of a twisted ring. In particular, the latter disturbance can cause vortex bursting in the early transitional stage. The impact of vortex bursting on the transition process is characterised by the near-wall, wall-normal velocity with the rapid distortion theory. The wall-normal velocity grows during vortex bursting, and leads to streak formation and then the transition to turbulence. The notable wall-normal velocity is induced by the large di-vorticity generated in vortex bursting. We model the growing radial component of the di-vorticity in terms of the local twist, and demonstrate that its surge is due to the generation of highly twisted vortex lines in vortex bursting. Then, we derive that the generation of the di-vorticity in the outer layer enhances the wall-normal velocity in the inner layer via the Poisson equation with the image method and the multipole expansion. Thus, we elucidate that the vortex bursting can have an effect on the transition process.

Snap-through is a buckling instability that allows slender objects, including those in plant and biological systems, to generate rapid motion that would be impossible if they were to use their internal forces exclusively. In microfluidic devices, such as micromechanical switches and pumps, this phenomenon has practical applications for manipulating fluids at small scales. The onset of this elastic instability often drives the surrounding fluid into motion – a process known as snap-induced flow. To analyse the complex dynamics resulting from the interaction between a sheet and a fluid, we develop a prototypical model of a thin sheet that is compressed between the two sides of a closed channel filled with an inviscid fluid. At first, the sheet bends towards the upstream direction and the system is at rest. However, once the pressure difference in the channel exceeds a critical value, the sheet snaps to the opposite side and drives the fluid dynamics. We formulate an analytical model that combines the elasticity of thin sheets with the hydrodynamics of inviscid fluids to explore how external pressure differences, material properties and geometric factors influence the system's behaviour. To analyse the early stages of the evolution, we perform a linear stability analysis and obtain the growth rate and the critical pressure difference for the onset of the instability. A weakly nonlinear analysis suggests that the system can exhibit a pressure spike in the vicinity of the inverted configuration.

We investigate theoretically and numerically the impact of an elastic sphere on a rigid wall in a viscous fluid. Our focus is on the dynamics of the contact, employing the soft lubrication model in which the sphere is separated from the wall by a thin liquid film. In the limit of large sphere inertia, the sphere bounces and the dynamics is close to the Hertz theory. Remarkably, the film thickness separating the sphere from the wall exhibits non-trivial self-similar properties that vary during the spreading and retraction phases. Leveraging these self-similar properties, we establish the energy budget and predict the coefficient of restitution for the sphere. The general framework derived here opens many perspectives to study the lubrication film in impact problems.

The classical paper by Lighthill (Commun. Pure Appl. Maths, vol. 109, 1952, p. 118) on the propulsion of ciliated microorganisms has become the reference against which many modern studies on swimming in low Reynolds number are compared. However, Lighthill's study was limited to propulsion in a uniform flow, whereas several biologically relevant microorganisms experience non-uniform flows. Here we propose a benchmark for ciliary propulsion in paraboloidal flows. We first consider the axisymmetric problem, with the microorganisms on the centreline of the background flow, and derive exact analytical solutions for the flow field. Our results reveal flow features, swimming characteristics and performance metrics markedly different from those generated in a uniform flow. In particular, the background paraboloidal flow introduces a Stokes quadrupole singularity at the leading-order flow field, generating vortices. Moreover, we determine the necessary conditions on the strength of the background flow for optimal power dissipation and swimming efficiency. We then consider the more general case of a microorganism off the centreline of the background flow. In this case, the squirmer experiences a paraboloidal, linear shear and uniform flows due to its position relative to the flow's centreline. Our findings show that while the linear shear flow does not affect the translational and rotational velocities of the squirmer, it does influence the velocity field and, therefore, the power dissipation.