We numerically investigate the propulsion of a two-dimensional compliant membrane executing prescribed harmonic heave in the near wake of a stationary circular cylinder at Reynolds number italic Re equals 3000 Re = 3000 $\textit{Re}=3000$ . Using a partitioned high-fidelity fluid–structure interaction solver with nonlinear iterative force correction, we map the coupled response over upper A Superscript asterisk Baseline element of left bracket 0.05 comma 0.5 right bracket A ∗ ∈ [ 0.05 , 0.5 ] $A^* \in [0.05,0.5]$ and f Superscript asterisk Baseline element of left bracket 0.1 comma 0.6 right bracket f ∗ ∈ [ 0.1 , 0.6 ] $f^* \in [0.1,0.6]$ , where upper A Superscript asterisk A ∗ $A^*$ is dimensionless flapping amplitude and f Superscript asterisk f ∗ $f^*$ is dimensionless flapping frequency. The parameter sweep reveals a sharp transition in the force maps from weakly forced, wake-following behaviour to a high-performance regime in which cycle-averaged lift and lift-to-drag increase abruptly. This transition coincides with intermittent exposure of the membrane to higher-momentum fluid and a pronounced amplification of deformation. A frequency-resolved analysis, combined with a body-frame Fourier mode decomposition of the flow, identifies four distinct flapping states, namely a wake-dominated state, two flapping-dominated states associated with lift reduction and lift gain and a two-way lock-in state. In the latter, vortex shedding locks onto the imposed actuation, while the first fluid-loaded structural mode approaches the actuation frequency, producing selective amplification of low-order deformation. The resulting curvature-induced camber intensifies leading-edge suction and increases the cross-membrane pressure difference, establishing feedback that reorganises upstream shedding. Motivated by these mechanisms, we derive scaling relations for the cycle-averaged lift, drag and power that separate quasi-steady motion, added-mass effects, curvature-induced contributions, wake-momentum deficit and transverse shear. For the present reference compliance, the scaling clarifies how flexibility is detrimental when the membrane remains fully immersed within the wake core, yet beneficial when intermittent wake exposure permits passive camber amplification to offset momentum deficit with limited drag penalty. These results provide a mechanistic framework for the present reference-compliance configuration and identify how wake exposure, imposed heaving and membrane response combine to produce two-way lock-in.
Rapidly rotating Rayleigh–Bénard convection on an f f $f$ -plane at colatitude theta Subscript f ϑ f $\vartheta _{\!f}$ is investigated numerically using an asymptotically reduced equation set valid in the limit of very rapid rotation. The equations provide a non-hydrostatic but quasi-geostrophic description in a non-orthogonal coordinate system. The tilt changes the structure of the large-scale barotropic condensate from large-scale vortices to zonal flows as the colatitude of the f f $f$ -plane increases, with bistable states present for certain parameter ranges, extending prior work to a geophysically significant parameter regime. This behaviour is understood through the impact of broken rotation symmetry on the barotropic source terms resulting from baroclinic vorticity stresses and baroclinic torque. As the tilt angle theta Subscript f ϑ f $\vartheta _{\!f}$ increases, global heat and momentum transport is reduced relative to upright-polar convection, a result that is explained through linear theory and nonlinear power maps both of which demonstrate increased attenuation of the domain of dynamically active spatial scales as the convective modes depart from a north–south alignment in the horizontal plane. A key finding is that the predominance of lateral thermal mixing allows for the maintenance of a persistent unstable mean temperature gradient that saturates at increasing forcing levels and remains insensitive to the colatitude.
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 upper Phi Subscript p comma 0 Baseline less than 1.0 times 10 Superscript negative 3 Φ p , 0 < 1.0 × 10 − 3 $\varPhi _{p,0}\lt 1.0\times 10^{-3}$ and italic St element of left parenthesis 0 comma 0.1 right parenthesis St ∈ ( 0 , 0.1 ) ${\textit{St}}\in (0,0.1)$ , respectively. The particle dynamics is governed by two critical Stokes numbers: the trapping critical Stokes number italic St Subscript italic cr comma t St cr,t ${\textit{St}}_{\textit{cr,t}}$ and the stability critical Stokes number italic St Subscript italic cr comma s St cr,s ${\textit{St}}_{\textit{cr,s}}$ . The critical Stokes number italic St Subscript italic cr comma t St cr,t ${\textit{St}}_{\textit{cr,t}}$ governs particle trapping. For italic St less than italic St Subscript italic cr comma t Baseline St < St cr,t ${\textit{St}} \lt {\textit{St}}_{\textit{cr,t}}$ , particles are trapped, forming a localised particle clustering ring (PCR) that induces substantial particle feedback on the flow. Furthermore, italic St Subscript italic cr comma s St cr,s ${\textit{St}}_{\textit{cr,s}}$ determines the stability of the PCR, which varies with upper Phi Subscript p comma 0 Φ p , 0 $\varPhi _{p,0}$ . A stable PCR forms when italic St less than italic St Subscript italic cr comma s Baseline St < St cr,s ${\textit{St}} \lt {\textit{St}}_{\textit{cr,s}}$ . At italic St Subscript italic cr comma s Baseline less than upper S t less than italic St Subscript italic cr comma t St cr,s < S t < St cr,t ${\textit{St}}_{\textit{cr,s}} \lt St \lt {\textit{St}}_{\textit{cr,t}}$ , 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 upper S Subscript italic max S max $S_{\textit{max}}$ and upper S 0 S 0 $S_0$ , 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.
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.
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 study combines linear stability analysis (LSA) and direct numerical simulations (DNS) to investigate the coupling mechanisms of rotation, stratification and shear in two flow geometries: spanwise rotating stratified plane Couette flow (between two infinite parallel plates) and sheared annular centrifugal Rayleigh–Bénard convection (between two concentric cylinders). First, LSA reveals a universal scaling law with respect to left parenthesis 1 minus eta right parenthesis ( 1 − η ) $(1-\eta )$ , where eta η $\eta$ denotes the radius ratio, for the deviation between annular and planar system results, quantitatively establishing the dynamical connection between annular and planar geometries. Building on this, DNS results indicate that the breakdown of the twin-force analogy in the turbulent regime originates from mode competition: buoyancy tends to drive three-dimensional chaotic plumes via baroclinic effects, whereas the Coriolis force tends to sustain quasi-two-dimensional longitudinal vortices. The Prandtl number determines the dominant mechanism in the near-wall region by regulating the relative thicknesses of the thermal and momentum boundary layers, thereby leading to a systematic dissimilarity between heat and momentum transport. Furthermore, this study provides a new theoretical basis for transport prediction in complex flows from the perspectives of enstrophy and mode competition.
Topographic roughness, defined as irregular variability of ocean depth on lateral scales of 1–10 km, has recently been shown to substantially impact the dynamics of large-scale and mesoscale ocean flows, including eddies, Rossby waves and baroclinic instability. This suggests that accurate modelling of roughness-induced forcing is critical for representing large-scale ocean behaviour. A recent multi-scale analysis of flows affected by small-scale bathymetry has produced the ‘sandpaper theory’, an explicit parametrisation of the roughness-induced bottom drag. This model accurately captures the impact of small-scale topography on large-scale flows without the need to resolve it in general circulation models. A peculiar facet of the sandpaper theory is that the drag is a non-monotonic function of the flow speed above the bottom boundary layer, which results in an instability. We characterise this non-monotonic drag instability for a series of barotropic and baroclinic systems, and explore a dynamically rich variety of scenarios that can produce stable jets, overstable Rossby waves, mesoscale eddy streets and self-amplifying hetons. We describe and quantify the properties of each flow type as they pertain to this non-monotonic drag regime.
Accurately measuring the instantaneous wall-shear stress in wall-bounded turbulent air flows is notoriously difficult. While conventional off-the-shelf flush-mounted hot-film sensors can be calibrated to acquire the time-averaged wall-shear stress, the instantaneous fluctuations are substantially underestimated as heat from the hot film is transferred into the substrate. To address this, nonlinear regression (NLR) is used to calibrate a flush-mounted hot film to enable accurate measurements of instantaneous wall-shear stress in turbulent boundary layers in a wind tunnel. During NLR calibration, the first four moments of wall-shear stress are acquired from either laser Doppler velocimetry (LDV), by linear fitting the streamwise velocity profile from within the viscous sublayer, or from direct numerical simulations matched to experiments by the Reynolds number based on the momentum thickness. The instantaneous wall-shear stress measured by the hot film is recovered with excellent agreement with that measured by LDV placed directly above the sensor. Direct numerical simulations show that the moments of wall-shear stress approach constant values for measurements taken below y Superscript plus Baseline equals 2 y + = 2 $y^+=2$ . Further investigation across different Reynolds numbers shows that all the second to fourth moments follow similar logarithmic trends. Inspired by this result, the sensor can be calibrated using only the first moment obtained in the same flow, while higher moments are estimated as functions of the Reynolds number. The versatility of the NLR approach is demonstrated through two case studies, showing that a posteriori NLR calibration can convert both new hot-film and historical near-wall hot-wire datasets into high-fidelity measurements of instantaneous wall-shear stress and reveal previously inaccessible physical signatures.
The deformation and breakup of drops in turbulent flows play a key role in a variety of industrial processes and natural situations. Literature reviews link the breakup of a droplet into two droplets (known as binary breakup) to a critical deformed state (CDS). Based on the droplet static pressure distribution, the CDS is defined as a level of deformation beyond which binary breakup is assumed to be inevitable and deterministic, i.e. even if no further external stresses act on the droplet. This study examines this CDS concept and identifies how sensitive rupture dynamics are to external stresses. These points are addressed using direct numerical simulations of individual droplets evolving in turbulence from their initially spherical state up to fragmentation. A scale-by-scale analysis is employed to describe droplet shape and deformation. It ensures an objective determination of the relevant scales involved in the CDS. The analysis of many simulated droplets shows that binary rupture is always preceded by the formation of a neck, whose narrowing until rupture is predicted by the CDS criterion of the literature. Supplementary simulations show that, depending on the shape of the droplets at the CDS, the turbulence has a negligible, contributory or essential effect on rupture. Thus, the binary breakup of droplets in a turbulent environment depends on the shape of the droplets and on their Ohnesorge number (ratio of viscous forces to capillary and inertial forces). These results suggest that the study should be supplemented with a comprehensive description of droplet morphology.
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 italic Ca Ca $ \textit{Ca}$ and viscosity ratio upper M M $M$ . A pore-network model predicts a sharp drop in growth rate at breakthrough and a geometry-controlled cross-over italic Ca Subscript times Ca × $ \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.
Nonlinear hydrodynamic effects influence the power capture of resonant wave energy converters (WECs), particularly within the frequency band where these devices are designed to operate. In this study, wave flume experiments are conducted on a semi-immersed spherical model with prescribed heave motions, emulating the response to an incident broad-banded wave group. Several body motion amplitudes (up to a maximum displacement of greater than half the radius) are tested at different peak frequencies, revealing progressive reductions in both the first-harmonic radiation forces and radiated wave elevations, per unit motion amplitude, as the motion amplitude is increased. A simple partially nonlinear potential flow model incorporating instantaneous ‘volume flow’ corrections shows very good agreement with the measured radiated waves, capturing the dominant nonlinear trends. An experimental decomposition of the radiated field into channel modes is also performed, based on linear theory while accounting for linear viscous dissipation. This is then used to calculate the power in the radiated field compared to the input power from the moving body over a frequency band of interest. The model and experiments show a consistent, monotonic decrease in normalised radiated power as the amplitude of body motion increases. Overall, the findings indicate that linear theory is remarkably accurate even for quite large body motions. The most important nonlinear corrections in the first-harmonic frequency range arise from third-order potential flow interactions, fundamentally caused by the spherical geometry, and demonstrate the potential utility of volume flow models for assessing WEC radiation.
We investigate the influence of insoluble surfactants on the spatio-temporal evolution of breaking waves, focusing on both regular and spilling regimes. Three-dimensional direct numerical simulations are conducted using an interface-tracking/level-set method that incorporates surfactant-induced Marangoni stresses. The simulations reveal that surfactant gradients, through Marangoni stresses, markedly alter the wave dynamics. While regular breakers exhibit only minor modifications in the presence of surfactants, increasing surfactant-induced Marangoni stresses in spilling breakers leads to changes in the crest evolution and vorticity generation. Surfactants enhance spilling breakers, primarily driven by Marangoni stresses rather than surface tension reduction. To quantify these effects, we also extend circulation-based theoretical frameworks to account for surfactant contributions.
Direct numerical simulations (DNS) are used to study the evolution of hairpin vortices to understand the late-stage turbulent breakdown of vortex rings. Several hairpin initial configurations with Reynolds number italic Re equals 1500 Re = 1500 $\textit{Re} = 1500$ were considered, both isolated and multiple. The isolated hairpins evolve in a quiescent flow, demonstrating reconnection events in stages (bridging, cut and reconnect, rapid separation of two parts, residual threads), similar to prior observations in antiparallel vortices. Next, multiple configurations of six hairpin vortices in a circular array are simulated, with the background flow extracted from the DNS of the near wake of a vortex ring undergoing azimuthal instability. Here, the alignment of vorticity at the tips with that of the azimuthal component of background flow vorticity is found to be an important parameter. When the vorticity components are opposite, hairpin tips deflect inward so that neighbouring hairpins also participate in reconnection events, yielding a complex set of reconnections, further amplified for staggered hairpins. Otherwise, hairpin tips deflect outwards and evolve much as an isolated hairpin. While higher italic Re Re $\textit{Re}$ makes the onset and reconnections faster, the extent of vortex stretching in the specific background flow relative to the viscous dissipation decides the temporal evolution of total enstrophy. It rises sharply just before reconnection events, falling rapidly afterwards, while for the DNS-extracted ring wake flow, the maximum enstrophy is higher compared with the simpler shear flows. Reconnection events distribute the energy over a broader spectrum, especially to higher wavenumbers.
Leveraging experimental measurements in rough-wall turbulent boundary layers that provided the spatial distribution and statistics of uniform momentum zones (UMZs), we previously introduced a stochastic model to generate two-dimensional modal velocity fields representative of the roughness sublayer and logarithmic region (Ehsani et al. 2024 a J. Fluid Mech. , vol. 979, p. A12; J. Fluid Mech. , 2024 b , vol. 999, p. A56). This synthetic flow comprises a streamwise concatenation of spatially correlated step-like velocity profiles, where each step corresponds to an internal shear layer separating UMZs, and the UMZs follow scaling consistent with wall-attached eddies. Here, we further develop the model to introduce small-scale swirling motions using vortex cores generated stochastically from previously published statistics and placed opportunistically with respect to the internal shear layers. Strategies are discussed to stretch and distribute vortices and extend the spatial resolution of the synthetic field to capture the smallest flow scales. The new synthetic fields for both laboratory and atmospheric settings reproduce with reasonable accuracy the second-order turbulence statistics and the energy spectrum for most of the inertial range, capturing, for the wind tunnel datasets, the transition to exponential decay up to italic k Subscript 1 Baseline eta tilde 0.2 k 1 η ∼ 0.2 $\textit{k}_{1}\eta \sim 0.2$ (where italic k Subscript 1 k 1 $\textit{k}_{1}$ is the streamwise wave number and eta η $\eta$ is the Kolmogorov length scale). The long-term goal is to extend and improve this bottom-up, statistically parameterised, reconstruction of rough-wall turbulence to open new wall modelling avenues for large eddy simulations.
We quantify the conversion of incident acoustic energy into vortical motion and viscous dissipation for a plane wave passing through large-aspect-ratio slit geometries. We perform direct numerical simulations over a broad imposed parameter space in incident sound pressure level (ISPL), Strouhal number ( italic St St $\textit{St}$ ) and Reynolds number ( italic Re Re $\textit{Re}$ ). Spectral proper orthogonal decomposition yields energy-ranked coherent structures at each frequency, from which we construct mode-by-mode fields for spectral kinetic energy (KE) and viscous loss (VL) to examine the acoustic absorption mechanisms. At italic ISPL equals 150 dB ISPL = 150 dB $\textit{ISPL} = {150}\,\textrm{dB}$ , the acoustic–hydrodynamic energy conversion is highest when italic St less than or equals 4 italic St Subscript 0 Baseline St ≤ 4 St 0 $\textit{St} \le 4\textit{St}_0$ , corresponding to an effective Keulegan–Carpenter number ( upper K Subscript c K c ${K_c}$ ) larger than 40. In this regime, three-dimensional simulations show that the dominant flow response is two-dimensional, with the oscillatory shear layer near the slit corners rolling up into shedding vortices. The VL accounts for 20 %–60 % of the KE contribution. For larger italic St St $\textit{St}$ , the Stokes-layer confinement produces X-shaped near-slit modes, reducing the energy input by approximately 50 %. The influence of italic Re Re $\textit{Re}$ depends on amplitude. Larger italic Re Re $\textit{Re}$ corresponds to suppressed broadband fluctuations and sharpened harmonic peaks at 150 dB 150 dB ${150}\,\textrm{dB}$ . At italic ISPL equals 120 dB ISPL = 120 dB $\textit{ISPL} = {120}\,\textrm{dB}$ , the boundary layers remain attached, vortex shedding is weak, absorption monotonically scales with viscosity and the italic Re Re $\textit{Re}$ - and italic St St $\textit{St}$ -dependencies become comparable. Across all conditions, more than 99 % of the VL is confined to a compact region surrounding the slit mouth. The KE–VL spectra identify regimes that enhance or suppress acoustic damping in slit geometries, providing a physically interpretable basis for acoustic-based design.
Predicting the pressure distribution in shock wave/boundary layer interaction (SWBLI) over convex surfaces is critical for hypersonic vehicle design, yet a unified theoretical model remains absent. This work extends the classical free-interaction theory by introducing two dimensionless parameters: normal upper Delta p overbar Subscript r Δ p ¯ r $\Delta \overline {p}_r$ , which quantifies the cumulative effect of downstream curvature on reattachment pressure, and normal upper Delta p overbar Subscript s Δ p ¯ s $\Delta \overline {p}_s$ , which captures the upstream displacement delay due to boundary-layer growth on the curved wall. Based on the relative position of the separation point with respect to the curvature onset, the interaction is classified into two distinct types: type-I, where separation occurs on the flat plate upstream of the curved section ( normal upper Delta p overbar Subscript s Baseline equals 0 Δ p ¯ s = 0 $\Delta \overline {p}_s = 0$ ), and type-II, where separation occurs on the curved surface itself ( normal upper Delta p overbar Subscript s Baseline less than 0 Δ p ¯ s < 0 $\Delta \overline {p}_s \lt 0$ ). Scaling methods are developed to reconstruct the entire pressure distribution within the separation region from the flat-plate baseline for both interaction types. Comprehensive validation using numerical simulations and experimental measurements demonstrates that the normalised pressure rise normal upper Delta upper F Subscript c Superscript asterisk divided by upper F Subscript p Superscript asterisk Δ F c ∗ / F p ∗ $\Delta F^*_c/F^*_p$ collapses linearly onto alpha Subscript r Baseline normal upper Delta p overbar Subscript r plus alpha Subscript s Baseline normal upper Delta p overbar Subscript s α r Δ p ¯ r + α s Δ p ¯ s $\alpha _r\,\Delta \overline {p}_r + \alpha _s\,\Delta \overline {p}_s$ with upper R squared greater than 0.98 R 2 > 0.98 $R^2 \gt 0.98$ , and the predicted overall pressure profiles agree excellently with numerical data. Empirical correlations for the model coefficients and a critical curvature radius are provided to define the applicability range. This work elucidates the physical mechanisms governing pressure distributions in curved-wall SWBLI, and establishes a systematic theoretical framework for pressure prediction in such configurations.
Advective dispersion of neutral solutes in flow is pertinent to the design of a host of physical systems, including microfluidic devices and chemical separation systems. The cross-sectional shapes of the channels within these devices are often complex and may vary (in shape or area) in the flow direction. However, previous analyses of Taylor–Aris dispersion in channels of varying geometry typically focus on either periodic channels or channels with simple cross-sections. To our knowledge, no reduced-order theory exists to predict dispersion in channels of arbitrary cross-section whose shape or size may vary slowly in the axial direction. In the current study, we develop a theoretical framework for a reduced-order model applicable to the analysis of dispersion in channels with such shapes given steady or time-dependent flow conditions. We first derive a one-dimensional partial differential equation for the area-averaged solute concentration and associated expressions for the full, three-dimensional tracer distribution. Next, we formulate two coupled ordinary differential equations that govern the solute axial mean position and variance. Additionally, we derive an ordinary differential equation to engineer channels to produce a priori specified solute variance distributions. We apply our model to the example geometries of axisymmetric channels of varying radii, channels with a varying ‘dumbbell’ cross-section and rectangular channels of varying width. We benchmark our model’s performance against Brownian dynamics simulations for all tested geometries. Overall, our framework is straightforward to apply to different geometries and flow conditions, and may yield critical insights for the design and optimisation of microfluidic systems.
We investigate the free fall of a flexible plate with a concentrated central weight using numerical simulations to explore the coupling effects of structural flexibility and concentrated loading. Two-dimensional and three-dimensional (3-D) simulations, combining lattice Boltzmann and immersed boundary methods, identify four distinct falling modes: swing, stable falling, fluttering and tumbling. The swing and stable falling modes are reported for the first time in this context. By analysing the balance between bending stiffness upper K K $K$ and concentrated weight upper G G $G$ , we introduce a new dimensionless effective stiffness italic Kg equals upper K divided by upper G Kg = K / G $ \textit{Kg} = K/G$ , which governs mode transitions and successfully collapses the kinematic data. A bistable transition zone between fluttering and tumbling is observed, with the mode selection dependent on the initial inclination angle, which scales as a power law of italic Kg Kg $ \textit{Kg}$ . Additionally, within the swing mode, we observe a transition in wake topology from a 2P to a 2S vortex-shedding pattern, modulated by the mass ratio. In 3-D simulations, the finite aspect ratio upper A Subscript r A r $A_r$ significantly influences mode selection due to pressure leakage at the side edges. While the italic Kg Kg $ \textit{Kg}$ scaling remains valid for the global dynamics, 3-D effects suppress tumbling in narrow plates and induce spanwise oscillations in the stable falling mode at higher aspect ratios.
This study examines the finger dynamics of high-amplitude, single-mode rarefaction-driven Rayleigh–Taylor instability (RTI) through rarefaction-tube experiments and numerical simulations. Unlike RTI driven by gravity or mechanical acceleration, rarefaction-driven RTI features a distinct wavefront-traversal (WT) stage prior to the linear stage. During the WT stage, finger velocities are decomposed into contributions from the rarefaction flow’s bulk motion and baroclinic-vorticity-induced velocity, and a theoretical framework is developed to quantitatively evaluate these components. Incorporating the WT-stage outcomes into the Layzer-type model (Li et al. 2025 J. Fluid Mech. 1016, A27) resolves the singularity-induced ill posedness at high amplitudes and, with the inclusion of interface stretching, enables accurate prediction of perturbation growth up to the quasi-steady stage. At high amplitudes, the inertial effect provides the fundamental contribution to bubble growth, whereas the curvature effect governs the transition to the quasi-steady state and the stretching effect determines the magnitude of the quasi-steady velocity. Beyond the quasi-steady stage, pronounced bubble deceleration emerges due to strong vorticity transport from the bubble region toward the spike. This transport reduces the interfacial vorticity that sustains bubble motion and thereby results in growth suppression. As the vortices shed from the spike reach the vicinity of the bubble tip, their induced velocity becomes significant, eventually terminating the deceleration and leading to a fluctuating saturated state.
