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.

While we now have a relatively good understanding of low-Reynolds-number hydrodynamics, and elegant techniques to dissect it, one cannot truly say the same for yield-stress fluids. For these materials, the nonlinearity associated with the yield stress complicates analysis and prevents the use of many of the techniques used for slow viscous flow. Simultaneously, the presence of a yield stress introduces a range of new features into the problem beyond those of traditional Stokes flow. Accordingly, in this essay, we discuss the impact of a yield stress in the relatively simple setting of two-dimensional, steady, inertialess flow. The main goals are to establish intuition for the dramatically different features that can be introduced to the flow by the yield stress, and to outline the various tools available to the modeller to construct and interpret these flows.

Direct numerical simulations are performed for turbulent forced convection in a half-channel flow with a wall oscillating either as a spanwise plane oscillation or to generate a streamwise travelling wave. The friction Reynolds number is fixed at $Re_{\tau _0} = 590$, but the Prandtl number $Pr$ is varied from 0.71 to 20. For $Pr\gt 1$, the heat transfer is reduced by more than the drag, 40 % compared with 30 % at $Pr=7.5$. This outcome is related to the different responses of the velocity and thermal fields to the Stokes layer. It is shown that the Stokes layer near the wall attenuates the large-scale energy of the turbulent heat flux and the turbulent shear stress, but amplifies their small-scale energy. At higher Prandtl numbers, the thinning of the conductive sublayer means that the energetic scales of the turbulent heat flux move closer to the wall, where they are exposed to a stronger Stokes layer production, increasing the contribution of the small-scale energy amplification. A predictive model is derived for the Reynolds and Prandtl number dependence of the heat-transfer reduction based on the scaling of the thermal statistics. The model agrees well with the computations for Prandtl numbers up to 20.

In gas evolving electrolysis, bubbles grow at electrodes due to a diffusive influx from oversaturation generated locally in the electrolyte by the electrode reaction. When considering electrodes of micrometre size resembling catalytic islands, direct numerical simulations show that bubbles may approach dynamic equilibrium states at which they neither grow nor shrink. These are found in under- and saturated bulk electrolytes during both pinning and expanding wetting regimes of the bubbles. The equilibrium is based on the balance of local influx near the bubble foot and global outflux. To identify the parameter regions of bubble growth, dissolution and dynamic equilibrium by analytical means, we extend the solution of Zhang & Lohse (2023) J. Fluid Mech. 975, R3, by taking into account modified gas fluxes across the bubble interface, that result from a non-uniform distribution of dissolved gas. The Damköhler numbers at equilibrium are found to range from small to intermediate values. Unlike pinned nano-bubbles studied earlier, for micrometre-sized bubbles the Laplace pressure plays only a minor role. With respect to the stability of the dynamic equilibrium states, we extend the methodology of Lohse & Zhang (2015a) Phys. Rev. E 91 (3), 031003(R), by additionally taking into account the electrode reaction. Under contact line pinning, the equilibrium states are found to be stable for flat nano-bubbles and for micro-bubbles in general. For unpinned bubbles, the equilibrium states are always stable. Finally, we draw conclusions on how to possibly enhance the efficiency of electrolysis.

Turbulent emulsions are ubiquitous in chemical engineering, food processing, pharmaceuticals and other fields. However, our experimental understanding of this area remains limited due to the multiscale nature of turbulent flow and the presence of extensive interfaces, which pose significant challenges to optical measurements. In this study, we address these challenges by precisely matching the refractive indices of the continuous and dispersed phases, enabling us to measure local velocity information at high volume fractions. The emulsion is generated in a turbulent Taylor–Couette flow, with velocity measured at two radial locations: near the inner cylinder (boundary layer) and in the middle gap (bulk region). Near the inner cylinder, the presence of droplets suppresses the emission of angular velocity plumes, which reduces the mean azimuthal velocity and its root mean squared fluctuation. The former effect leads to a higher angular velocity gradient in the boundary layer, resulting in greater global drag on the system. In the bulk region, although droplets suppress turbulence fluctuations, they enhance the cross-correlation between azimuthal and radial velocities, leaving the angular velocity flux contributed by the turbulent flow nearly unchanged. In both locations, droplets suppress turbulence at scales larger than the average droplet diameter and increase the intermittency of velocity increments. However, the effects of the droplets are more pronounced near the inner cylinder than in the bulk, likely because droplets fragment in the boundary layer but are less prone to break up in the bulk. Our study provides experimental insights into how dispersed droplets modulate global drag, coherent structures and the multiscale characteristics of turbulent flow.

Combined theoretical and quantitative experimental study of resonant internal standing waves in a pycnocline between two miscible liquids in a narrow rectangular basin is presented. The waves are excited by a cylinder that harmonically oscillates in the vertical direction. A linear theoretical model describing the internal wave structure that accounts for pycnocline thickness, the finite wavemaker size and dissipation is developed. Separate series of measurements were performed using shadowgraphy and time-resolved particle image velocimetry. Accurate density profile measurements were carried out to monitor the variation of the pycnocline parameters in the course of the experiments; these measurements were used as the input parameters for the model simulations. The detected broadening of the pycnocline is attributed mainly to the presence of the waves and leads to the variation of the wave structure. The complex spatio-temporal structure of the observed internal wavefield was elucidated by carrying band-pass filtering in the temporal domain. The experiments demonstrate the coexistence of multiple spatial modes at the forcing frequency as well as the presence of the internal wave system at the second harmonic of the forcing frequency. The results of the theoretical model are in good agreement with the experiments.