We solve a Bayesian inverse Navier–Stokes (N–S) problem that assimilates velocimetry data by jointly reconstructing a flow field and learning its unknown N–S parameters. We devise an algorithm that learns the most likely parameters of a Carreau shear-thinning viscosity model, and estimates their uncertainties, from velocimetry data of a shear-thinning fluid. We conduct a magnetic resonance velocimetry experiment to obtain velocimetry data of an axisymmetric laminar jet in an idealised medical device (US Food and Drug Administration’s benchmark nozzle) for a blood analogue fluid. The algorithm successfully reconstructs the flow field and learns the most likely Carreau parameters. Predictions from the learned model agree well with rheometry measurements. The algorithm accepts any differentiable algebraic viscosity model, and can be extended to more complicated non-Newtonian fluids (e.g. Oldroyd-B fluid if a viscoelastic model is incorporated).

A linear stability analysis of a soluble surfactant-laden liquid film flowing down a compliant substrate is performed. Our purpose is to expand the prior studies (Carpenter and Garrad 1985 J. Fluid Mech. 155, 465–510; Alexander et al., 2020 J. Fluid Mech. 900, A40) by incorporating a soluble surfactant into the flow configuration. As a result, we formulate the Orr–Sommerfeld-type boundary value problem and solve it analytically by using the long-wave series expansion as well as numerically by using the Chebyshev spectral collocation method in an arbitrary wavenumber regime for infinitesimal disturbances. The long-wave result reveals that surface instability is stabilized in the presence of a surfactant, whereas it is destabilized in the presence of a compliant substrate. These opposing impacts suggest an analytical relationship between parameters associated with the soluble surfactant and compliant wall, ensuring the same critical Reynolds number for the emergence of surface instability corresponding to both surfactant-laden film flow over a compliant wall and surfactant-free film flow over a non-compliant wall. In the arbitrary wavenumber regime, along with the surface mode, we identify two additional modes based on their distinct phase speeds. Specifically, the wall mode emerges in the finite wavenumber regime, while the shear mode emerges only when the Reynolds number is large. As the surfactant Marangoni number increases, the wall mode destabilizes, resulting in a different outcome from the surface mode. Moreover, increasing the value of the ratio of adsorption and desorption rate constants stabilizes surface instability but destabilizes wall mode instability. As a result, we perceive that the soluble surfactant-laden film flow is linearly more unstable than the insoluble one due to surface instability but linearly more stable than the insoluble one due to wall mode instability. Additionally, we see a peculiar behaviour of base surface surfactant concentration on the primary instability. In fact, it has a specific value depending on adsorption and desorption rate constants below which surface instability stabilizes but wall mode instability destabilizes, whereas above which an opposite phenomenon occurs. Finally, in the high-Reynolds-number regime, we can suppress shear mode instability by raising the surfactant Marangoni number and the ratio of adsorption and desorption rate constants when the angle of inclination is sufficiently small. Unlike surface instability, the base surface surfactant concentration exhibits both stabilizing and destabilizing influences on shear mode instability.

Spontaneous flow reversals in buoyancy-driven flows are ubiquitous in many fields of science and engineering, often characterized by violent, intermittent occurrences. In this study, we present a complex-network-based reduced-order model to analyse intermittent events in turbulent flows, using temporal and spatial snapshot data. This framework combines elements of dynamical system theory with network science. We demonstrate its utility by applying it to data sequences from intermittent flow reversal events in two-dimensional thermal convection. This approach has proven robust in detecting and quantifying structures and predicting reversals. Additionally, it provides a perspective on the physical mechanisms underlying flow reversals through cluster evolution. This purely data-driven methodology shows the potential to enhance our understanding, prediction and control of turbulent flows and complex systems.

The wake systems of ducted and conventional marine propellers are compared for a highly loaded condition by exploiting results of large eddy simulations, conducted on a cylindrical grid consisting of 3.5 billion points. The results demonstrate a dramatic change of both performance and flow physics, due to the nozzle. The efficiency of propulsion is increased by about $30\,\%$, but the thrust generated by the propeller is reduced, replaced in most part by that produced by its nozzle. As a result, weaker coherent structures are shed in the wake on the ducted propeller, compared with the conventional one. Meanwhile, the tip leakage vortices experience a faster instability into smaller turbulent structures. Therefore, the wake signature of the ducted propeller, detrimental to its interaction with downstream bodies, is reduced, compared with that of the conventional propeller operating with no duct. The source of the faster instability of the tip leakage vortices is different from the typical one of the tip vortices shed by conventional propellers. The latter is attributable to phenomena of short- and long-wave instabilities of the helices of each tip vortex, eventually leading to mutual inductance, leapfrogging and breakup into turbulence. In contrast, the former is tied to the shear developed between the tip leakage vortices and the boundary layer of the inner surface of the nozzle, rather than to the interaction between vortices shed by different blades.

We investigate through numerical simulations the hydrodynamic interactions between two rigid spherical particles suspended on the axis of a cylindrical tube filled with an elastoviscoplastic fluid subjected to pressure-driven flow. The simulations are performed by the finite-element method with the arbitrary Lagrangian–Eulerian formulation. We carry out a parametric analysis to examine the impact of the yield stress and relaxation time of the fluid and of particle confinement on the dynamics of the system. We identify master curves of the particle relative velocity as a function of the inter-particle distance. When the yield stress of the suspending phase is much lower than the viscous stress, those curves highlight short-range attractive interactions and long-range repulsive interactions between particles, with the latter specifically promoting their alignment. As the yield stress increases, the attractive interaction is replaced by stasis at short distance, characterised by a vanishing relative velocity and the formation of an unyielded region that connects the two spheres, where the fluid behaves like a viscoelastic solid. Additionally, the combined effects of plasticity and elasticity enhance the repulsion between the particles, promoting their ordering. Also, increasing the confinement of the particles enhances repulsion, thus allowing us to achieve ordering within shorter lengths in the flow direction. Reducing shear thinning amplifies peak relative velocities and expands the attractive region due to increased viscoelastic stresses and stress gradients. While a stable equilibrium may appear at larger separations, its impact is limited by low relative velocities.

Direct numerical simulations of the injection of a pulsed round liquid jet in a stagnant gas are performed in a series of runs of geometrically progressing resolution. The Reynolds and Weber numbers and the density ratio are sufficiently large for reaching a complex high-speed atomisation regime but not so large so that the small length scales of the flow are impossible to resolve, except for a very small liquid-sheet thickness. The Weber number based on grid size is then small, an indication that the simulations are very well resolved. Computations are performed using octree adaptive mesh refinement with a finite volume method and height-function computation of curvature, down to a specified minimum grid size $\varDelta$. Qualitative analysis of the flow and its topology reveals a complex structure of ligaments, sheets, droplets and bubbles that evolve and interact through impacts, ligament breakup, sheet rupture and engulfment of air bubbles in the liquid. A rich gallery of images of entangled structures is produced. Most processes occurring in this type of atomisation are reproduced in detail, except at the instant of thin sheet perforation or breakup. We analyse droplet statistics, showing that as the grid resolution is increased, the small-scale part of the distribution does not converge, and contains a large number of droplets close in order of magnitude to the minimum grid size with a significant peak at $d = 3\varDelta$. This non-convergence arises from the numerical sheet breakup effect, in which the interface becomes rough just before it breaks. The rough appearance of the interface is associated with a high-wavenumber oscillation of the curvature. To recover convergence, we apply the controlled ‘manifold death’ numerical procedure, in which thin sheets are detected, and then pierced by fiat before they reach a set critical thickness $h_c$ that is always larger than $6 \varDelta$. This allows convergence of the droplet frequency above a certain critical diameter $d_c$, above and close to $h_c$. A unimodal distribution is observed in the converged range. The number of holes pierced in the sheet is a free parameter in the manifold death procedure; however, we use the Kibble–Zurek theory to predict the number of holes expected on heuristic physical grounds.

The propagation of sound waves in high-temperature and plasma flows is subject to attenuation phenomena that alter both the wave amplitude and speed. This finite change in acoustic wave properties causes ambiguity in the definition of sound speed travelling through a chemically reactive medium. This paper proposes a novel computational study to address such a dependence of sound-wave propagation on non-equilibrium mechanisms. The methodology presented shows that the equations governing the space and time evolution of a small disturbance around an equilibrium state can be formulated as a generalised eigenvalue problem. The solution to this problem defines the wave structure of the flow and provides a rigorous definition of the speed of sound for a non-equilibrium flow along with its absorption coefficient. The method is applied to a two-temperature plasma evolving downstream of a shock, modelled using Park’s two-temperature model with 11 species for air. The numerical absorption coefficient at low temperatures shows excellent agreement with classical theory. At high temperatures, the model is validated for nitrogen and argon across wide temperature ranges with experimental values, showing that classical absorption theory is insufficient to characterise high-temperature flows due to the effect of finite-rate chemistry and vibrational relaxation. The speed of sound is verified in the frozen and equilibrium limits and its non-equilibrium profile is presented with and without viscous effects. It is furthermore shown that the variation in the speed of sound is driven by the dominating reaction mechanisms that the flow is subject to at different thermodynamic conditions.

Developing a consistent near-wall turbulence model remains an unsolved problem. The machine learning method has the potential to become the workhorse for turbulence modelling. However, the learned model suffers from limited generalisability, especially for flows without similarity laws (e.g. separated flows). In this work, we propose a knowledge-integrated additive (KIA) learning approach for learning wall models in large-eddy simulations. The proposed approach integrates the knowledge in the simplified thin-boundary-layer equation with a data-driven forcing term for the non-equilibrium effects induced by pressure gradients and flow separations. The capability learned from each flow dataset is encapsulated using basis functions with the corresponding weights approximated using neural networks. The fusion of capabilities learned from various datasets is enabled using a distance function, in a way that the learned capability is preserved and the generalisability to other cases is allowed. The additive learning capability is demonstrated via training the model sequentially using the data of the flow with pressure gradient but no separation, and the separated flow data. The capability of the learned model to preserve previously learned capabilities is tested using turbulent channel flow cases. The periodic hill and the 2-D Gaussian bump cases showcase the generalisability of the model to flows with different surface curvatures and different Reynolds numbers. Good agreements with the references are obtained for all the test cases.

Shallow cumuli are cloud towers that extend a few kilometres above the atmospheric boundary layer without significant precipitation. We present a novel laboratory experiment, boiling stratified flow, as an analogy to study turbulent mixing processes in the boundary layer by shallow cumulus convection. In the experimental beaker, a syrup layer (representing the atmospheric boundary layer) is placed below a freshwater layer (representing the free troposphere) and heated from below. The temperature is analogous to the water vapour mixing ratio in the atmosphere, while the freshwater concentration is analogous to the potential temperature. When the syrup layer starts boiling, bubbles and their accompanying vortex rings stir the two-layer interface and bring colder fresh water into the syrup layer. Two distinct regimes are identified: transient and steady boiling. If the syrup layer is initially sufficiently thin and diluted, then the vortex rings entrain more cold water than needed to quench superheating in the syrup layer, ending the boiling. If the syrup layer is initially deep and concentrated, then the boiling is steady since the entrainment is weak, causing the entrained colder water to continuously prevent superheating. A theory is derived to predict the entrainment rate and the transition between the two regimes, validated by experimental data. Finally, analogies and differences with the atmospheric processes are discussed.

This paper investigates the linear and nonlinear dynamics of two-dimensional penetrative convection subjected to radiative volumetric thermal forcing, focusing on ice-covered freshwater systems. Linear stability analysis reveals how critical wavenumbers $k_c$ and Rayleigh numbers $Ra_c$ are influenced by the attenuation lengths and incoming heat flux. In this configuration, the system easily becomes unstable with a small $Ra_c$, which is two decades smaller than that of the classical Rayleigh–Bénard convection problem, with typically $O(10)$. Weakly nonlinear analysis figures out that this configuration is supercritical, contrasting with the subcritical case by Veronis (Astrophys. J., vol. 137, 1963, 641–663). Numerical bifurcation solutions are performed from the critical points and over several decades, up to $Ra \sim O(10^6)$. This paper found that the system exhibits multiple steady solutions, and under certain specific conditions, a staircase temperature profile emerges. Meanwhile, we further discuss the influence of incoming heat flux and the Prandtl number $Pr$ on the primary bifurcation. Direct numerical simulations are also carried out, showing that heat is transported more efficiently via unsteady convection.

In inviscid, incompressible flows, the evolution of vorticity is exactly equivalent to that of an infinitesimal material line-element and, hence, vorticity can be traced forward or backward in time in a Lagrangian fashion. This elegant and powerful description is not possible in viscous flows due to the action of diffusion. Instead, a stochastic Lagrangian interpretation is required and was recently introduced, where the origin of vorticity at a point is traced back in time as an expectation over the contribution from stochastic trajectories. We herein introduce for the first time an Eulerian, adjoint-based approach to quantify the back-in-time origin of vorticity in viscous, incompressible flows. The adjoint variable encodes the advection, tilting and stretching of the earlier-in-time vorticity that ultimately leads to the target value. Precisely, the adjoint vorticity is the volume-density of the mean Lagrangian deformation of the earlier vorticity. The formulation can also account for the injection of vorticity into the domain at solid boundaries. We demonstrate the mathematical equivalence of the adjoint approach and the stochastic Lagrangian approach. We then provide an example from turbulent channel flow, where we analyse the origin of high streamwise wall-shear-stress events and relate them to Lighthill’s mechanism of stretching of near-wall vorticity.

At constant pressure, a mixture of water parcels with equal density but differing salinity and temperature will be denser than the parent water parcels. This is known as cabbeling and is a consequence of the nonlinear equation of state for seawater density. With a source of turbulent vertical mixing, cabbeling has the potential to trigger and drive convection in gravitationally stable water columns and there is observational evidence that this process shapes the thermohaline structure of high-latitude oceans. However, the evolution and maintenance of turbulent mixing due to cabbeling has not been fully explored. Here, we use turbulence-resolving direct numerical simulations to investigate cabbeling’s impact on vertical mixing and pathways of energy in closed systems. We find that cabbeling can sustain convection in an initially gravitationally stable two-layer configuration where relatively cold/fresh water sits atop warm/salty water. We show the mixture of the cold/fresh and warm/salty water is constrained by a density maximum and that cabbeling enhances mixing rates by four orders of magnitude. Cabbeling’s effect is amplified as the static stability limit is approached, leading to convection being sustained for longer. We find that available potential energy, which is classically thought to only decrease with mixing, can increase with mixing due to cabbeling’s densification of the mixed water. Our direct numerical dimulations support the notion that cabbeling could be a source of enhanced ocean mixing and that conventional definitions of energetic pathways may need to be reconsidered to take into account densification under mixing.

This study examines the effect of nozzle flexibility on vortex ring formation at a Reynolds Number of Re = 1000. The flexible nozzles impart elastic energy to the flow, increasing the hydrodynamic impulse of the vortex ring dependent on the input fluid acceleration and the initial nozzle tip deflection (predicted by the measured nozzle damped natural frequency). When these time scales are synchronised, the output velocity and hydrodynamic impulse of the vortex ring are maximised. Vortex ring pinch-off is predicted using the output velocity for each nozzle and is confirmed with closed finite time Lypunov exponent contours. The lowest tested input formation length, L/D = 1, where L is the piston stroke length and D is the nozzle diameter, generates a greater increase in impulse than L/D = 2 and L/D = 4, due to a higher relative increase in total ejected volume and by remaining in the single vortex formation regime. At L/D = 2 and L/D = 4, multiple vortex structures are observed due to the interplay of the counter-flow generated by the nozzles re-expanding and the steady input flow. At the end of the pumping cycle, during fluid deceleration, the flexible nozzles collapse. This helps in suppressing unfavourable negative pressure regions from forming within the nozzle, instead expelling additional fluid from the nozzle. Upon reopening, beneficial stopping vortices form within the nozzles, with circulation correlated to nozzle stiffness. This highlights a secondary optimal stiffness criterion that must be considered in a full-cycle analysis: the nozzle must be compliant enough to collapse during deceleration, yet remain as stiff as possible to reopen quickly to maximise efficiency in refilling.

In recent years, various unique properties of microswimmer suspensions have been revealed. Some microswimmers are deformable; however, the influence of the swimmer’s deformability has been overlooked. The present study examined the impact of soft microswimmers’ membrane deformations in a mono-dispersed dense suspension on microstructure formation. Due to the small size of the microswimmers, the flow field is described by the Stokes equation. The soft microswimmer was modelled as a capsule with a two-dimensional hyperelastic membrane enclosing a Newtonian fluid that is driven by propulsion torques distributed slightly above the membrane surface. Changes to the torque distribution caused the soft swimmer to exhibit different swimming modes as a pusher or puller. Similar to rigid squirmers, soft swimmers displayed self-organised local clusters in the suspension. Membrane deformation changed the mutual interference among swimmers in the cluster, bringing the interactions closer together than those of rigid squirmers. Especially among soft pushers, rotational diffusion due to hydrodynamic interference was reduced and the swimming trajectory became relatively straight. As a result, polar order was less likely to form, especially in regions of high $Ca$. On the other hand, pullers showed strong interactions due to retraction flow and an increase in mean membrane tension. For pushers (pullers), the rear (side) interaction produced the greatest change in tension. These findings are expected to be useful for effort to understand the propulsion mechanisms of medical and industrial soft microrobots, as well as the biological responses of microorganisms induced by mechanical stimuli.

The system composed of a circular cylinder free to move along a transverse rectilinear path within a cross-current has often served as a canonical problem to study the vortex-induced vibrations (VIV) developing in the absence of structural restoring force, thus without structural natural frequency. The object of the present work is to extend the exploration of the behaviour of this system when the path is set to an arbitrary orientation, varying from the transverse to the streamwise direction, and the cylinder is forced to rotate about its axis. The investigation is conducted numerically at a Reynolds number equal to $100$, based on the body diameter and oncoming flow velocity, for structure to displaced fluid mass ratios down to $0.01$ and values of the rotation rate (ratio between body surface and oncoming flow velocities) ranging from $0$ to $1$. When the transverse symmetry is broken by the orientation of the trajectory or the forced rotation, the cylinder drifts along the rectilinear path, at a velocity that can be predicted by a quasi-steady approach. Three distinct regimes are encountered: a pure drift regime, where the body translates at a constant velocity, and two oscillatory regimes, characterised by contrasted forms of displacement fluctuation about the drifting motion, but both closely connected to flow unsteadiness. VIV, nearly sinusoidal, persist over a wide range of path orientations, for all rotation rates. On the other hand, irregular jumps of the body, triggered by the rotation and named saccades, emerge when the trajectory is aligned, or almost aligned, with the current. The two forms of response differ by their regularity, but also by their amplitudes and frequencies, which deviate by one or more orders of magnitude. The rotation attenuates both VIV and saccades. Yet, an increase of the rotation rate enhances the erratic nature of the saccade regime.

We investigate the dynamics of close-contact melting (CCM) on ‘gas-trapped’ hydrophobic surfaces, with specific focus on the effects of geometrical confinement and the liquid–air meniscus below the liquid film. By employing dual-series and perturbation methods under the assumption of small meniscus deflections, we obtain numerical solutions for the effective slip lengths associated with velocity $\lambda$ and temperature $\lambda _t$ fields, across various values of aspect ratio $\Lambda$ (defined as the ratio of the film thickness $h$ to the structure’s periodic length $l$) and gas–liquid fraction $\phi$. Asymptotic solutions of $\lambda$ and $\lambda _t$ for $\Lambda \ll 1$ and $\Lambda \gg 1$ are derived and summarised for different surface structures, interface shapes and $\Lambda$, which reveal a different trend of $\lambda$ for $\Lambda \ll 1$ and depending on the presence of a meniscus. In the context of constant-pressure CCM, our results indicate that longitudinal grooves can enhance heat transfer under the effects of confinement and a meniscus when $\Lambda \lesssim 0.1$ and $\phi \lt 1 - 0.5^{2/3} \approx 0.37$. For gravity-driven CCM, the parameters of $l$ and $\phi$ determine whether the melting rate is enhanced, reduced or nearly unaffected. We construct a phase diagram based on the parameter matrix $(\log _{10} l, \phi )$ to delineate these three regimes. Lastly, we derive two asymptotic solutions for predicting the variation in time of the unmelted solid height.

In this study, we investigate the sedimentation of spheroidal particles in an initially quiescent fluid by means of particle-resolved direct numerical simulations. Settling particles with three different shapes – oblate spheroid, sphere and prolate spheroid – but fixed Galileo number $Ga=80$ and density ratio $\gamma =2$ at volume fraction $\phi =1\%$ are considered. Oblate and prolate particles are found to form column-like clusters as a consequence of the wake-induced hydrodynamic interactions in the suspension. This effect, together with the change of particle orientation, enhances the mean settling velocity of the dispersed phase. In contrast, spherical particles do not exhibit clustering, and settle with hindered velocity in the suspension. Furthermore, we focus on the pseudo-turbulence induced by the settling particles. We report a non-Gaussian distribution of the fluid velocity and a robust $-3$ power law of the energy spectra. By scrutinizing the scale-by-scale budget, we find that the anisotropy of the particle-induced pseudo-turbulence is manifested not only by the uneven allocation of turbulence kinetic energy among the different velocity components, but also by the anisotropic distribution of energy in spectral space. The fluid–particle interactions inject energy into the vertical velocity component, thus sustaining the turbulence, while pressure redistributes the kinetic energy among the different velocity components. The clustering of oblate/prolate particles significantly increases the energy input at large scales, forcing elongated flow structures. Moreover, the redistribution and nonlinear transfer of the energy are also intensified in the presence of particle clustering, which reduces the anisotropy of the particle-induced pseudo-turbulence.

This study employs volume-of-fluid-based computational fluid dynamics modelling to investigate the coupled effects of surface wettability and inflow vapour velocity on R134a ($p/p_{cri}=0.25$) condensation heat transfer in horizontal tubes. The results demonstrate that both the condensation heat transfer coefficient (HTC) and Nusselt number consistently increase with rising vapour velocity, indicating enhanced convective heat transfer at higher flow rates. Within this overall trend, the influence of surface wettability varies significantly across different velocity regimes. At moderate inlet velocities (10 m s−1), surface wettability demonstrates maximum impact, with the HTC enhancement exceeding 19.1% between peak and minimum values, optimising at contact angles of 120$^\circ$–140$^\circ$. As velocity increases to 20 m s−1, while surface wettability effects persist with $\gt$11.7 % enhancement, convective heat transfer becomes increasingly dominant, showing $\gt$38.8 % improvement in the maximum HTC compared with the 10 m s−1 case. At higher velocities (40 m s−1), the influence of surface wettability diminishes substantially, with the HTC variation reducing to $\gt$1.04 %. At extreme velocities (80 m s−1), surface tension effects become negligible compared with vapour shear forces, resulting in minimal (0.53 %) variation across different contact angles. The equivalent Reynolds number peaks at 20 m s−1, indicating optimal conditions for condensate formation and flow characteristics. These findings provide crucial insights for condensation system design, suggesting that while increasing velocity generally enhances heat transfer performance, surface wettability modifications are most effective at moderate velocities, while high-velocity applications should prioritise flow dynamics and system geometry optimisation.