When a less viscous fluid displaces a more viscous one in a porous medium, the interface becomes unstable, forming viscous fingers. While viscous fingering has been studied in depth, previous work has focused on the pre-breakthrough dynamics – that is, before the injected fluid reaches any outlet boundary. Here, we investigate post-breakthrough behaviour in a radial micromodel through experiments and pore-network simulations. Post-breakthrough displacement is key for practical applications, as it governs how much of the defending fluid is ultimately recovered. We show that the displacement pattern either ‘freezes’ or continues to grow, depending on the capillary number $ \textit{Ca}$ and viscosity ratio $M$. A pore-network model predicts a sharp drop in growth rate at breakthrough and a geometry-controlled cross-over $ \textit{Ca}_\times$ below which the displacement pattern is arrested. These results highlight the role of domain size, pore geometry, fluid properties and injection rate in regulating the post-breakthrough dynamics and provide a framework for optimising displacement efficiency.

Integral identities relating the skin-friction coefficients and the Reynolds shear stresses in annular cylindrical flows are derived. In the pressure-driven case, an identity for the radial location of the maximum streamwise velocity is obtained and, in the cylinder-driven case, an identity for the bulk velocity is found. The formulas are used to analyse existing numerical and experimental data and simplify to classical channel-flow and pipe-flow identities in the limiting cases of vanishingly small and infinitely large radius ratios.

We study by means of direct numerical simulations the influence of the dimensionality of convection on flow properties. We call attention to a few general principles from considering in totality the results from one dimension, two dimensions, three dimensions and four dimensions. In particular, we explore two practical aspects: (i) the transient time, or the amount of time it takes for the flow to reach the steady state; and (ii) possible implications for the so-called ultimate state.

This study experimentally investigates the interfacial instabilities and thermo-hydrodynamic mechanisms of supercavities ventilated by high-temperature condensable vapour. Using high-speed imaging with an AI-based segmentation model, we establish a flow regime map that delineates the supercavitation envelope, identifying a distinct rough supercavity regime associated with the interaction between phase change and turbulence. A key finding is the non-monotonic cavity-length response of condensable vapour ventilation, in which high-flow regression is governed by a condensation-assisted source–sink imbalance, and differs mechanistically from the leakage/closure response observed in air ventilation. Specifically, condensation-induced mass loss acts as an internal sink that can exceed the vapour supply and cause cavity regression. While the cavity diameter remains inertially constrained, its length is highly sensitive to these thermodynamic effects. Spatiotemporal analysis indicates a coupled thermo-hydrodynamic route for the interfacial instability. The observed instability is initiated by thermodynamic self-excitation driven by droplet impingement, under-expanded jet expansion, and contact condensation within the ventilation influence region. These disturbances are convected downstream through a Kelvin–Helmholtz-type shear-layer process. The upper–lower asymmetry of the cavity fluctuations is consistent with gravity-related selection, which preferentially amplifies long-wave perturbations on the upper interface through a Rayleigh–Taylor-type route, and damps fluctuations on the lower interface. These findings extend classical cavitation theory into regimes dominated by rapid phase change, providing new insights into thermo-hydrodynamic coupling in multiphase flows.

We present direct numerical simulations of two-dimensional vertical convection (VC) to investigate how the aspect ratio $A$ influences the evolution of flow states and heat transport. The study covers a Rayleigh number range of $10^{7}\le \textit{Ra}_H \le 10^{11}$ at a fixed Prandtl number of $\textit{Pr}=1$ with $A$ varying in the range of $1/50\le A \le 8$. It is found that, for a fixed value of $A$, the flow evolves from steady convection to periodic oscillations, chaotic flows and finally turbulence as $\textit{Ra}_H$ increases. At fixed $\textit{Ra}_H$, increasing $A$ induces a similar sequence of transitions, demonstrating that geometric confinement plays an important role in determining the dynamical state of the system. A flow regime diagram in the $(\textit{Ra}_H, A)$ space shows that the critical Rayleigh numbers for these transitions all follow approximately $\textit{Ra}_{H,c}\sim A^{-3}$. The flow develops a stably stratified bulk, whose normalised mean vertical temperature gradient $S$ varies non-monotonically with $A$. The geometric confinement also significantly affects the transport. The Nusselt number $\textit{Nu}_H$ varies non-monotonically with $A$ and reaches a maximum at an optimal aspect ratio $A_{opt}\sim \textit{Ra}_H^{-0.21}$. In contrast, the Reynolds number $Re_H$ increases monotonically with $A$ and exhibits a crossover in its $\textit{Ra}_H$ scaling at $A_c\sim \textit{Ra}_H^{-0.22}$. Finally, we demonstrate that the plate height $L$ provides a more appropriate length scale. Using $L$-based non-dimensionalisation collapses the transport data across a wide range of $A$ onto universal scaling laws, i.e. $\textit{Nu}_L=0.16\textit{Ra}_L^{0.28}$ and $\textit{Re}_L=0.30\textit{Ra}_L^{0.49}$, and renders the transition thresholds $\textit{Ra}_{L,c}$ nearly independent of $A$. This universality indicates that the transport and flow destabilisation are governed primarily by the development of the boundary layers along the vertical walls in VC.

Supercritical fluid flows exhibit complex dynamics and stability characteristics because of drastic thermophysical property variations across the pseudo-boiling line. Conventional stability analysis methods, primarily developed for ideal gases at low pressures, cannot be directly applied to supercritical flows. The present study establishes a unified framework of global linear stability analysis (GLSA) suitable for supercritical fluid flows for the first time. The framework incorporates real-fluid effects in terms of a cubic equation of state and detailed treatment of thermodynamic and transport properties, and integrates both energy and species transport equations into the linearised governing equations. For demonstration, the developed GLSA framework is implemented to examine the stability characteristics of jets in crossflow at different supercritical pressures and momentum flux ratios ($J$). Global linear stability analysis provides quantitative assessments of growth rates of dynamical events by recognising large eigenvalues and associated eigenmodes. The results show that the real-fluid modification is critical to accurately capture the growth rate magnitude of unstable modes. At low supercritical pressure, the position of the dominant GLSA eigenmode shifts from near the nozzle exit to farther downstream, accompanied by a significant reduction in growth rate magnitude. This shift is in alignment with the transition from global to convective instability in the jet upstream shear layer, where the Rayleigh–Taylor instability is enhanced owing to a pronounced density gradient. Furthermore, the GLSA eigenvalue spectrum quantifies the distinction in stability characteristics induced by the real-fluid effect even under identical $J$. The novel real-fluid algorithm can be extended to the stability analysis of many other supercritical fluid flows.

Low-friction-Reynolds-number ($ \textit{Re}_\tau \approx 235 {-} 300$ and $ \textit{Re}_\tau \approx 400 {-} 480$) turbulent boundary layers (TBLs) perturbed by small, mesoscale and relatively large synthetic streamwise vortices are investigated using planar particle image velocimetry (PIV) measurements and wall-resolved large-eddy simulation (LES). To better understand how these varying size streamwise vortices evolve in TBLs, synthetic streamwise vortex pairs are generated from a family of geometrically self-similar miniature vortex generators of viscous-scaled heights $h^+ \approx 24, 48, 96$ in experiments and $h^+ \approx 24, 30, 48$ in simulations. Analyses employing a triple decomposition of velocity found that the wall-normal domains of influence of these vortices are distinct and show dependence on the wall-normal peak location of the primary Reynolds shear stress component. It is found that the evolution of the synthetic vortices in TBLs is scale-dependent and characterised by the interplay between the positive (momentum source) and negative (momentum sink) components of the turbulent inertia, and additional inertia induced by the vortices. The ratio of the gradient of the viscous stress (of a canonical TBL) to the turbulent inertia acts as a diagnostic for estimating the wall-normal domain of influence of the synthetic vortices. Analyses of wall-normal domains of influence suggest that synthetic vortices of sufficient size ‘escape’ from the near-wall region. This scale effect on the evolution of embedded streamwise vortices is closely connected to the spread of time-invariant spatial fluctuations associated with secondary flows in TBLs. Estimation of the ‘escape’ scale provides insights into how the near-field mean dynamics of these vortices is related to their resulting dynamical signatures in the far field.

Vortex-induced vibration forever (VIVF) is classically associated with a single elastically mounted cylinder, for which oscillations persist at infinite reduced velocity when the mass ratio ($m^*$) falls below a critical value ($m_{cr}^*$). The existence and characteristics of such permanent vibration under unsteady wake conditions remain unclear. This study investigates the vibration response of an extreme-light cylinder placed in the wake of an upstream fixed cylinder without restoring forces, with particular emphasis on the existence and variation of a critical mass ratio $m_{cr}^*$ with the spacing ratio ($L/D$). The Reynolds number is set to 60, the mass ratio $m^*$ ranges from 0.2 to 14 and the spacing ratio $L/D$ ranges from 1.1 to 6.0. Three vibration regimes are identified in the $(L/D,\, m^*)$ parameter space: non-vibration (NV), limited-range vibration (LRV) and infinite-range vibration (IRV). A critical mass ratio $m_{cr}^*$ emerges at the LRV–NV boundary, where the vibration amplitude continuously diminishes to zero as $m^*$ approaches $m_{cr}^*$, in contrast to the abrupt transition observed in single-cylinder VIVF; this extinction coincides with the disappearance of the second-order flow mode. Furthermore, $m_{cr}^*$ is governed by the spacing ratio relative to the vortex formation length $L_{\!f}/D$ of a stationary cylinder: when $L/D \lt L_{\!f}/D$, $m_{cr}^*$ increases with $L/D$, whereas for $L/D \gt L_{\!f}/D$, it decreases with further increase in $L/D$. In contrast, within the IRV regime the flow maintains persistent co-shedding behaviour and no critical mass ratio is observed within the investigated $m^*$ range, indicating that wake interference reshapes the stability boundary of the system.

Direct numerical simulations are performed to investigate the effects of discontinuous slip boundaries comprising alternating slip regions and transverse no-slip stripes on turbulent channel flow and drag reduction mechanisms. Compared with a continuous slip boundary with the same effective slip length, the discontinuous configuration yields a substantially lower net drag reduction. This degradation is attributed to a diminished slip contribution resulting from the triple effects of slip area deficit, localised flow-field re-equilibration and inter-stripe interference, coupled with variations in the turbulence-modulation contribution and an intensified dispersive penalty. The no-slip stripes generate local adverse pressure gradients and induce shear-layer instabilities, triggering the development of internal boundary layers at the leading and trailing edges. This boundary-layer re-equilibration process manifests as a characteristic ‘oscillation-overshoot-decay recovery’ in the wall velocity gradient. Simultaneously, the no-slip stripes enhance turbulent kinetic energy production in the buffer layer and intensify wall-normal transport. While variations in the no-slip stripe geometries mainly affect drag reduction through the slip component and become dynamically important when the pitch $\lambda$ is sufficiently small, slip anisotropy offers a more robust mitigation strategy. By suppressing spanwise slip, the lateral motions amplified by no-slip stripes are effectively blocked, rendering the boundary layer insensitive to stripe-induced localised disturbances. This mechanism enables streamwise-slip-dominant discontinuous boundaries to achieve drag reduction performance that significantly surpasses that of isotropic or spanwise-slip-dominant configurations.

We investigate the role of particle wettability in interfacial trapping by oil drops in a turbulent oil–water emulsion, using interface-resolved numerical simulations that account for turbulence, interface deformation, and particle–interface interactions. Using this framework, we characterise the trapping efficiency of hydrophilic, neutrally wetting and lipophilic particles. The oil–water interface is described by a phase-field method, while particle motion is resolved through a Lagrangian approach that accounts for both interfacial adhesion and wettability effects, enabling the direct simulation of asymmetric particle capture under turbulent conditions. Our results show a pronounced asymmetry in trapping efficiency with respect to neutral wetting, with a maximum attained for lipophilic particles. This asymmetry originates from the wettability-controlled equilibrium configuration of trapped particles: lipophilic particles preferentially reside on the drop side of the oil–water interface, where turbulent stresses are significantly weaker. As a result, these particles exhibit reduced detachment rates, leading to enhanced trapping efficiency compared to neutrally wetting and hydrophilic particles. Finally, we propose a minimal first-order kinetic model, whose rate coefficients are extracted from the simulations, that accurately predicts trapping efficiency as a function of particle wettability.

In this work we propose a physics-constrained machine learning approach to enhance the generalisability of neural-network-represented turbulence models across different flow regimes. Machine learning-augmented turbulence modelling has emerged to improve the predictive accuracy of the Reynolds-averaged Navier–Stokes method based on high-fidelity data. However, such data-driven models face difficulties in generalising well in different flow scenarios outside training datasets. Here we show that this issue can be alleviated by incorporating physical constraints of the Reynolds stress during the learning process. It is achieved with a physics-constrained learning approach based on the regularised ensemble Kalman method, which allows for model learning from limited velocity data and is flexible in imposing physical constraints. The method is used to learn a nonlinear eddy viscosity model, represented by a tensor-basis neural network, from velocity data in separated flows over periodic hills. Two physical constraints are incorporated during the model learning, i.e. the baseline coefficients to capture the log law of the wall and the algebraic Reynolds stress equation to inform the Reynolds stress anisotropy. Moreover, an adaptive penalty method is developed to enforce physical constraints in the feature space associated with canonical flows, avoiding the incompatibility with the training data from separated flows. Our results show that the proposed method can learn a generalisable turbulence model across different flow scenarios, including similar separated flows, canonical wall-bounded flows and complex flow configurations. Finally, the learned neural-network-based model function is discussed, demonstrating its consistency with the imposed physical constraints.

Swimming microorganisms dynamically respond to their external environment, resulting in self-organised collective behaviour. To accurately predict the flow field at the continuum scale, it is essential to understand the rheological properties of active suspensions. Microorganisms in nature exhibit a wide range of deformabilities, and former studies have reported that they play an important role in hydrodynamic interactions. However, the effect of cell deformability on suspension rheology has not yet been clarified. In this study, we developed a soft microswimmer model to represent general swimming microorganisms to investigate rheological properties such as apparent viscosity and normal stress differences of their suspensions. The model consisted of a capsule with a hyperelastic membrane and propelled itself by a torque distribution modelling the ciliary activity of a ciliate. Our results demonstrated that interparticle hydrodynamic interaction is a critical factor inducing symmetry breaking in the time-averaged orientational distribution in the low shear flow. For the ellipsoidal swimmers, these interactions promote orientational distribution biased towards the elongational direction, resulting in inherent viscosity variation in microbial suspensions. In addition, this orientational distribution is distinctly different from that observed in solitary swimmers, causing the significant reduction (increase) in the apparent viscosity for the pusher (puller). Analysis varying membrane deformability showed that the softer membrane is less likely to exhibit the effects of the active stress on specific viscosity. These findings demonstrate the important role of elastohydrodynamic interactions between swimmers in suspension rheology and support the development of continuum models for complex microbial systems.

We study experimentally the time evolution of energy and its spectrum in decaying quasi-two-dimensional (quasi-2-D) turbulence with different damping rates $\gamma$. The thin-layer fluid flow is forced electromagnetically and decays freely after cessation of the forcing. We found that energy does not exhibit an exponential decay associated with linear damping, but an algebraic decay which can be described by a quadratic drag model. This represents the first experimental evidence of the quadratic drag in (quasi-) 2-D turbulence. The energy spectrum $E(k)$ undergoes two decay stages. In the initial stage, the shape of $E(k)$ changes with time and is notably dependent on $\gamma$. In the late stage, $E(k)$ evolves in a self-preserving manner and exhibits an unexpected universal form that is independent of $\gamma$. This universal spectrum is characterised by a dual scaling: a $k^{-3}$ scaling at large scales and $k^{-5/3}$ at small scales, exhibiting remarkable similarity with the Nastrom–Gage spectrum in atmospheric flows. The $k^{-3}$ scaling is found to be associated with an inverse energy transfer, while $k^{-5/3}$ may be attributed to the enhancement of forward energy transfers after the removal of body force that suppresses the three-dimensional motions at small scales. The profile of inverse energy flux is shifted to larger scales during decay, resulting in further growth of turbulent structures.

The dynamics of Lagrangian particles dispersed in rotating turbulence differ fundamentally from those observed in classical homogeneous isotropic turbulence (HIT). Rotation induces a pronounced two dimensionalisation of the carrier flow, governed by inertial waves and coherent columnar vortices, while particles experience additional Coriolis and centrifugal accelerations that compete with the standard Stokes drag. In this study, high-fidelity direct numerical simulations of HIT and rotating turbulence in a triply periodic domain are used to investigate the clustering and Lagrangian dynamics of millions of sub-Kolmogorov point particles. Increasing rotation reorganises both the flow and the dispersed phase: moderate rotation produces a domain-scale columnar vortex that expels inertial particles via drag-Coriolis coupling, while further increasing rotational effects amplifies centrifugal segregation and promotes anisotropic organisation, ultimately leading to the formation of slender, particle columns representative of a limiting clustering regime. Rotation significantly enhances particle clustering and anisotropy, shifting the maximum preferential concentration toward larger Stokes numbers. These trends are quantified using a new diagnostic based on the gyration tensor, which provides a unified description of cluster concentration, morphology and orientation across scales. To link these static signatures to their dynamical origin, the geometry and acceleration of Lagrangian trajectories are analysed. Increasing rotation suppresses vertical acceleration component while amplifying the planar one, steering particle dispersion toward an effectively two-dimensional regime. This transition is reflected in curvature and torsion statistics, which reveal more strongly bent trajectories and a reduced decorrelation from large-scale vortical structures.

Multiscale vortex structures significantly influence inertial particle dynamics, contributing to clustering in turbulent flows – a phenomenon common in many applications. In this study, our primary objective is to investigate how particle clustering is influenced by vortex effects while accounting for particle feedback. Using a two-way coupled numerical method, we identify novel particle dynamics characteristics, as localised particle clustering triggers periodic leakage, even in a dilute particle-laden unequal-strength counter-rotating vortex pair (CVP). Small, heavy, dilute inertial particles are considered. In particular, the initial particle volume fraction and the particle Stokes number are set to $\varPhi _{p,0}\lt 1.0\times 10^{-3}$ and ${\textit{St}}\in (0,0.1)$, respectively. The particle dynamics is governed by two critical Stokes numbers: the trapping critical Stokes number ${\textit{St}}_{\textit{cr,t}}$ and the stability critical Stokes number ${\textit{St}}_{\textit{cr,s}}$. The critical Stokes number ${\textit{St}}_{\textit{cr,t}}$ governs particle trapping. For ${\textit{St}} \lt {\textit{St}}_{\textit{cr,t}}$, particles are trapped, forming a localised particle clustering ring (PCR) that induces substantial particle feedback on the flow. Furthermore, ${\textit{St}}_{\textit{cr,s}}$ determines the stability of the PCR, which varies with $\varPhi _{p,0}$. A stable PCR forms when ${\textit{St}} \lt {\textit{St}}_{\textit{cr,s}}$. At ${\textit{St}}_{\textit{cr,s}} \lt St \lt {\textit{St}}_{\textit{cr,t}}$, a PCR forms and then periodic temporal leakage of particles occurs. The significant feedback from the PCR creates a morphologically similar positive vorticity region (PVR) through the baroclinic source. The periodic transport of vorticity within the PVR modulates the critical streamlines $S_{\textit{max}}$ and $S_0$, which play decisive roles in particle trapping. These modulations directly reduce the stable region of the PCR and displace its location, enabling particles to escape.

Mathematical modelling of the microcirculatory haemodynamics in the retina is an essential tool for understanding various diseases of the retina, yet remains challenging due to the multiscale nature of the retinal vasculature and its coupling to surrounding tissue. To address this, we develop a multiscale model that couples retinal vasculature across scales with interstitial tissue. Our model combines the one-dimensional model for arterioles and venules with the coupled Darcy equations for capillaries and tissue. The model uses an analytic solution for the capillary–tissue coupled system that provides a simple interpretation of the results along with much faster computation. The analytic solution implies a dynamic coupling condition that links the capillary bed with upstream arteriolar and downstream venular flows. The model is mathematically robust, demonstrated through analysis of the solution’s truncation error and convergence. Its predictive accuracy is validated against experimental data and other models, making it useful in interpreting experimental results. Finally, the role of various parameters in controlling retinal haemodynamics is explored.

Direct numerical simulations of the flow over a compound delta-wing model reveal a novel receptivity mechanism for the second-mode instability in hypersonic boundary layers. This mechanism involves disturbances that propagate obliquely upward from the wall, interact with the shock wave and reflect back into the boundary layer. Detailed analysis indicates that this process strongly modulates high-frequency instabilities, such as the second mode, while exerting minimal influence on low-frequency cross-flow and first-mode disturbances. Upon re-entry into the boundary layer, the reflected disturbances cause rapid, stepwise amplification of the internal second mode, demonstrating that the mechanism modulates and enhances boundary-layer instability. Further analysis shows that a phase-locked mechanism is involved in these interactions. The identification of this receptivity mechanism offers new insights into hypersonic boundary-layer transition.

Acoustic liners are passive sound-absorbing materials widely used in engineering applications and are commonly characterised by their acoustic impedance. In the presence of high sound pressure levels or grazing flows, the impedance is modified by local nonlinear effects. Conventional impedance measurements are typically obtained through frequency-domain eduction techniques, which infer impedance from its effect on measurable acoustic quantities and therefore rely on assumptions regarding wave propagation and near-wall acoustic–flow interactions. In this paper, a time-domain impedance eduction approach based on instantaneous wall-normal velocity measurements is developed to investigate the nonlinear response of acoustic liners under grazing-flow conditions. Unlike conventional methods, the proposed approach does not rely on a wave-propagation model and reconstructs the liner impedance directly from the local velocity field. The method is assessed using both laser Doppler velocimetry measurements and high-fidelity numerical simulations. The resulting impedance estimates show good agreement with classical predictions while revealing that a significant part of the flow effect originates from turbulence-induced wall-normal velocity fluctuations acting through the same nonlinear mechanisms as high-amplitude acoustic excitation. Furthermore, the commonly reported upstream–downstream impedance mismatch largely disappears when wave-propagation modelling is removed from the eduction process, suggesting that this discrepancy primarily arises from modelling assumptions rather than from an intrinsic dependence of the liner impedance on the direction of acoustic propagation.

This work investigates how spanwise twisting modulates aerodynamic performance and leading-edge vortex (LEV) dynamics of a low aspect-ratio (AR) flapping wing through variations in effective angle of attack ($\alpha _{\textit{eff}}$). Spanwise twisting primarily redistributes force between the streamwise and spanwise directions for a low-AR flapping wing – reallocating force orientation rather than altering total force magnitude – thereby directly governing the thrust-to-resultant-force ratio and hence propulsive efficiency. At low Strouhal number, excessive $\alpha _{\textit{eff}}$ induces stall, while at medium-to-high Strouhal number, resultant force remains insensitive to $\alpha _{\textit{eff}}$ as increasing twist enhances thrust at the expense of side force. Maximum propulsive efficiency of 0.36 occurs at Strouhal number = 0.6 and $\alpha _{\textit{eff,max}} = 27^{\circ }$, where the resultant force is optimally oriented in the streamwise direction, maximising the thrust-to-resultant-force ratio. Circulation analysis reveals that peak efficiency corresponds to a circulation plateau during late downstroke, sustained by a secondary structure, a lambda-shaped positive vortical structure beneath the LEV, that regulates the LEV growth. This mechanism is absent at both excessively low and high $\alpha _{\textit{eff,max}}$. Spanwise vortex variations indicate functional differentiation: inboard regions maintain attached flow for lift, while outboard regions exploit LEV dynamics for thrust. Moreover, late twisting in a flapping cycle enhances efficiency by promoting vortex shedding before the stroke reversal, whereas early twisting in a flapping cycle amplifies thrust.

We address the non-equilibrium structure of triple points, the region of confluence of three shocks and a shear layer arising in gas dynamics. The structure of triple points and their reflections in diatomic gases was computed in two dimensions using a diatomic extension of the 10-moment model. This approximate Boltzmann model allows for the accurate modelling of the relaxation process of translational and rotational temperatures within the shock structure and shear layers. The initial flow field was prescribed based on idealised three-shock theory, and the interaction between the triple-point and a plane of symmetry was studied. While the large-scale dynamics of triple-shock reflections was found to be in excellent agreement with the Euler model, the non-equilibrium calculation permitted us to establish the unique structure of the triple-point region. We discovered the presence of an internal shock inside the triple point, oriented normal to the flow. Its internal interaction with the Mach shock is responsible for the presence of an internal hot spot, generating temperatures significantly exceeding those expected from the treatment of each shock as discontinuities. We also provide a detailed appraisal of principal translational temperatures associated with these internal hot spots. The important role of this temperature anisotropy inside triple points on cellular detonations is briefly discussed.