Papers

Continuous release is a fundamental source condition in transport, yet theoretical treatments have focused on downstream concentration development for passive solutes or fully developed cross-sectional distributions of active particles. We develop a spatial theory for the steady transport of active particles under continuous release in confined shear flows, resolving the concentration field from the inlet to the far field. Based on the Smoluchowski equation, we formulate a boundary-value problem with point-source inlet flux and boundary conditions. Separation of the streamwise coordinate from the cross-sectional variables yields a non-self-adjoint generalized eigenvalue problem, whose spatial modes are superposed to construct the solution. The eigenproblem is solved within a Galerkin spectral framework; downstream-admissible modes are retained and their coefficients determined from the inlet flux through a weighted biorthogonal expansion. Excellent agreement with individual-based simulations validates the theory. For plane Poiseuille flow under convection-dominated conditions, we find pronounced position–orientation coherence for spherical particles in the early developing region: swimming and shear reorientation drive the population through successive angular stages, generating alternating off-centre and centreline accumulation regions downstream, while dispersion progressively weakens this coherence. Particle elongation enhances orientational alignment, producing an early three-peaked vertical profile and stronger, more persistent lateral accumulation downstream, while reducing the coherence of centreward migration and thereby suppressing secondary centreline accumulation. The streamwise marginal concentration varies non-monotonically, reflecting changes in the mean streamwise velocity, with its far-field limit given by the reciprocal of the asymptotic drift velocity.

Purcell's swimmer is a well-known planar model of a swimming microorganism, governed by low Reynolds number hydrodynamics, which is comprised of three rigid links connected by actuated rotary joints. This model has been analyzed as a robotic locomotion system governed by first-order nonlinear dynamics with a periodic input (gait) of the two joint angles. In this work, we present a robotic macro-scale realization of this three-link swimmer moving in a highly viscous fluid. We propose a simple variant of Purcell's theoretical model with non-slender links and a central rigid sphere which represents the added drag of the robot's central flotation block, and calibrate the model's parameters to fit experimental measurements. Next, we apply optimal control formulation based on Pontryagin's Maximum Principle (PMP) in order to find optimal gaits that maximize the displacement per cycle under bounds on the joint angles. Employing a differential geometric method that transforms the problem to area integral enclosed by the gait trajectory in the plane of joint angles, enables visual interpretation which explains topological changes in displacement-optimal gaits upon varying the bound on the joint angles. We then apply PMP formulation to the problem of maximizing Lighthill's energy efficiency in order to obtain a boundary value problem (BVP) whose solution gives efficiency-optimal gaits for Purcell's swimmer model, as well as its variant with a central sphere. Finally, we utilize numerical methods such as parameterizing the input gait as a truncated Fourier series, as well as GPOPS-II solver, to produce sufficient initial guess values for solving the BVPs and obtaining efficiency-optimal gaits.

Time-resolved direct simulation Monte Carlo (DSMC) fields are used to test whether a detached rarefied hypersonic bow shock contains a slow collective displacement that can be separated from correlated particle-sampling fluctuations. Mach-10 rotationally relaxing nitrogen flow over a circular cylinder is analysed for diameter-based Knudsen number $0.01\leq \KnD\leq1$, where $\KnD=\lambda_\infty/D$, $\lambda_\infty$ is the freestream mean free path and $D$ is the cylinder diameter. A density half-jump front is extracted on body-normal rays, unsupported solid-side points are excluded, and temporal coarse graining is performed before feature extraction. Persistent and sampling covariance components are compared using a penalized composite-fit score, design-scale cross-validation, block resampling, synthetic controls and complementary full-field matched filters. Corrected field proper orthogonal decomposition (POD) is high rank at every Knudsen number, yet a weak, same-signed angular displacement is resolved at $\KnD=0.01$ and $0.025$. Independent random-seed and simulator-particle-loading repeats recover the angular shape and relaxation time while the raw sampling variance changes with loading. Across the two resolved states the mean density layer broadens by $82\%$, while the angular shapes remain strongly aligned. Density and pressure recover the marker motion most strongly; the reduced Mach-number and translational-temperature participation at $\KnD=0.025$ is evidence consistent with moment-selective weakening, although observable-dependent signal-to-noise remains a possible contributor. The signal is interpreted as a low-pass bow-layer response embedded in broadband kinetic fluctuations, not as a newly discovered discrete oscillation or a demonstrated linear instability. The higher-Knudsen records are not sufficiently sensitive to establish physical disappearance.

We present a sharp and conservative geometrical VOF–finite-volume method for multicomponent liquid–gas mass transfer across deformable interfaces. The method solves problems in which multiple species are coupled through interfacial mass balances, latent-heat exchange, and vapor–liquid equilibrium. The key novelty is a fully sharp two-field treatment of scalar transport: the species and temperature equations are solved separately in the liquid and gas phases, while the one-sided Robin conditions for species and the two-sided flux jump for temperature are imposed directly on the reconstructed interface through an embedded-boundary discretization. This avoids both volumetric regularization of interfacial source terms and explicit coupling based on previous-time-step interfacial data. A consistent geometrical advection scheme is used for volume, momentum, energy, and species transport, and a sequential coupling strategy is developed to determine the partial interfacial mass fluxes, close the temperature equation, and update the thermodynamic-equilibrium state. The method is validated through single- and multicomponent bubble dissolution, single-component droplet evaporation, non-ideal ethanol–isooctane droplet evaporation, and sessile water–glycerol droplet evaporation. The results demonstrate second-order accuracy, accurate interfacial flux prediction, good mass and energy conservation, and the ability to capture complex multicomponent effects such as gas replacement, azeotropic volatility reversal, and composition-driven Marangoni flow.

Liquid bridges suspended between two parallel horizontal cylinders are studied using experiments, reduced-order mathematical modelling and numerical simulations. Both non-electrified and electrified configurations are considered, with the cylinders acting as electrodes between which a potential difference can be applied. The initial focus is on equilibrium bridge shapes, while the dynamics is examined only in the non-electrified case. In the experiments, transformer-oil bridges are investigated and modelled as perfect dielectrics. The electric field counteracts the gravitational effects and causes the bridges to rise and become flatter, with shapes that are well described by Young–Laplace-type equations containing non-local electric-field contributions evaluated using a boundary-element method. Bifurcation diagrams of these solutions are constructed by pseudo-arclength continuation to characterise the dependence of bridge shapes on liquid volume and electric-field strength. In the absence of an electric field, the transient relaxation to equilibrium is analysed using a reduced-order model developed using Onsager's variational principle, with comparison to direct numerical simulations. The steady states predicted by the reduced-order model agree closely with those of the full formulation over a wide parameter range, and the dynamics is well captured in the overdamped regime.

A body floating atop a supersaturated fluid may accumulate bubbles along its underbelly, which can render the body rotationally unstable. But rotation can strip the surface of these bubbles when they make contact with the air above. To explore this coupling we perform experiments using cylinders floating on carbonated water. The cylinders exhibit an array of distinct dynamics, including constant rolling, periodic and aperiodic oscillation, chaos, and intermittent capsizing, which can be tuned by body size, mass distribution, and gas concentration. A continuum model for the bubble density dynamics, and Galerkin projection, tie the experiment to the generalized Lorenz system and its famed chaotic attractor. Even an extremely small center-of-mass offset can have a qualitative impact on the dynamics, and an offset of as little as a few percent can fully stabilize the system. The experiment represents a highly accessible physical realization of the Lorenz system, which, owing to continuous gas loss from the fluid to the air above, sweeps slowly without intervention across a classical bifurcation diagram in time.

Turbulence modeling in porous media can be greatly improved by combining high-resolution numerical methods with modern data-driven techniques. The development of accurate macroscale models (length scale greater than the pore size) will enable real-time systemic simulations of porous media flow. We consider the case of turbulent flow in homogeneous porous media, typically encountered in engineered porous media (heat exchangers, metamaterials, combustors, etc.). The underlying microscale flow field is inhomogeneous and determined by the geometry of the porous medium. Neural Networks are able to resolve the geometry-dependence and the non-linearity of porous media turbulent flow. We are proposing to separate the macroscale model into individual blocks that predict a unique aspect of the microscale flow, such as microscale spatial flow distribution and vortex dynamics. In the present work, we determine the feasibility of the prediction of the Reynolds-averaged microscale flow patterns by using Convolutional Neural Networks (CNN). The porous medium is represented by using a square lattice arrangement of circular cylinder solid obstacles. The pore-scale Reynolds number of the flow is 300. The porosity of the porous medium is varied from 0.45 to 0.92 with 60 steps. The microscale flow field is simulated by using Large Eddy Simulation (LES) with a compact sixth-order finite difference method. We demonstrate satisfactory prediction of the microscale flow field using the CNN with a global error less than 10%. We vary the number of training samples to study the deterioration of the model accuracy. The CNN model offers a O(106) speedup over LES with only 10% loss in accuracy.

Snow is a complex geomaterial whose macroscopic response is governed by density, temperature, and the topology of its evolving microstructure. Its mechanical behavior spans elastic, plastic, viscous, and failure dominated regimes, imposing significant challenges for numerical methods, which intends to simulate large deformations, evolving free surfaces, and complex boundary interactions. This work presents the first integration of a Modified Cam-Clay constitutive formulation for snow into a purely meshfree strong-form collocation framework based on the Generalized Finite Difference Method. The main methodological contribution is a numerical coupling that combines a global implicit mixed formulation for pressure and velocity with a constitutive return-mapping algorithm. The hydrostatic pressure contribution is obtained from a Poisson equation and subsequently corrected through the Modified Cam-Clay return-mapping procedure, while the deviatoric response is treated semi-implicitly using a numerical viscosity formulation. This partitioned treatment of the volumetric and deviatoric stress contributions enables stable simulations with comparatively large time steps while producing smooth spatial pressure fields. As a result, forces on complex boundary geometries can be evaluated accurately. Numerical results of this coupling illustrate the algorithmic stability of the framework, the effective imposition of boundary conditions, and the suitability of local spatial refinement. The feasibility of applying the framework to vehicle-snow interaction through rigid-body coupling is also illustrated. The presented formulation provides a robust basis for future simulations of dynamic snow loading on vehicle structures.

The generalised Lagrangian mean (GLM) theory of Andrews & McIntyre provides a powerful framework to study the interactions between waves and flows. A drawback of this theory is that the Lagrangian mean velocity is divergent even for incompressible fluids because the mean flow map, which sends the Lagrangian labels of fluid parcels to their mean positions, does not preserve volume. This results, for instance, in vortices shrinking under Lagrangian averaging. We overcome this drawback by revising the definition of the mean flow map, choosing it as the volume-preserving map closest to the "bare" GLM mean map. A standard result of optimal-transport theory then shows that the new mean map is the volume-preserving factor in the polar factorization of the GLM mean map. We develop and implement a numerical method for the computation of the corresponding Lagrangian mean fields from simulation data. The implementation builds on recently developed algorithms for the on-the-fly computation of Lagrangian means using the exponential and Butterworth filters. We demonstrate the value of volume-preserving Lagrangian averaging in simulations of the two-dimensional incompressible and shallow-water models. We compare the Lagrangian-mean fields obtained with and without the volume-preservation constraint.

In this work, a novel phase-field method is proposed for the six- and seven-equation non-equilibrium models for simulating compressible two-phase flows. Such formulations allow for monotonic mixture speed of sound, minimizing artificial wave delay during transmission across an interface. The proposed phase field formulation is constructed from the baseline seven-equation model, and interface-regularization terms are added in divergence form, while maintaining consistency between the partial differential equations without introducing spurious source terms. It admits conservative phasic and mixture entropy transport equations, thus facilitating the construction of discrete conservative schemes. The six-equation formulation is obtained under instantaneous velocity equilibrium. To avoid eigenvector degeneracy of the system of PDEs, the volumetric interface regularization flux is modified to account for a finite amount of conjugate phase, which improves on how phasic density is captured implicitly. Stability of compressible two-phase flow rely on the preservation of the interface-equilibrium conditions, and the preservation of discrete kinetic energy and entropy. A detailed analysis of IEC demonstrates additional requirements on the consistency of flux splittings between the convective and interface-regularization terms for all quantities, as well as the effects on the phasic internal energy flux splittings. A KEEP discretization is proposed and evaluated over a suite of high-density ratio test cases, including interface advection, acoustic wave-induced bubble oscillation, oblique acoustic wave reflection and transmission, and two-phase Taylor-Green vortex flow. Results demonstrate accuracy, stability and robustness for very long time integrations, a desired feature for simulation of turbulent flows and acoustics, since the framework does not rely on the addition of numerical dissipation.

Wall-pressure fluctuations beneath a turbulent boundary layer drive the flow-induced noise and structural loading of rough surfaces. On a smooth wall their variance grows with the logarithm of the Reynolds number, a growth carried, in a source-term reading, by the nonlinear turbulence–turbulence part of the pressure source acting across the logarithmic layer. We ask what sets the same growth once the wall is fully rough. Standard rough-wall phenomenology answers it: above the roughness the mean flow keeps its smooth-wall logarithmic form, so the active source range is cut off at the roughness height rather than the viscous length, and the variance grows on the logarithmic span between the roughness height and the layer thickness in place of the Reynolds number. Splitting the pressure source into its two physical parts then distinguishes separate contributions to the energy: a roughness-local one from the mean-shear source, fixed at high frequency on the roughness scale, and an energetic one from the nonlinear source at the outer scale, which alone carries the growth. Calibrated against rough-wall cases, the model collapses the spectral shape onto this energetic peak and holds it Reynolds-independent at fixed geometry. The canopy contribution decays with distance from the wall and contributes a finite offset. The growth coefficient is predicted to be the smooth-wall one. The present range is too narrow to discriminate that rate from twice or half it, so the contribution is a parameter-free prediction and the identification of a suitable roughness-size sweep that would test it.

Inclined gravity currents on shallow slopes can exhibit pronounced turbulence intermittency. Using direct numerical simulations, we investigate this behaviour for a temporal gravity current over a range of initial Reynolds numbers $Re_0$. For $Re_0=2500$ and a slope angle of $0.5^\circ$, the outer layer of the current exhibits large excursions in turbulence intensity and repeated transitions between turbulent and weakly turbulent states. Analysis of the flow energetics reveals that the intermittency is associated with a finite delay between shear production and dissipation of turbulent kinetic energy. During transitional phases, this delay permits a transient amplification of turbulence, which significantly weakens the mean shear by extracting kinetic energy from the mean flow and promoting entrainment-driven layer growth, ultimately leading to relaminarisation. Increasing $Re_0$ reduces the delay and progressively suppresses intermittency, steering the flow towards a more sustained turbulent state. Motivated by these observations, we develop an autonomous delay-differential model based on the coupled evolution of the mean and turbulent kinetic energies. The model reproduces the observed transition from intermittent to sustained turbulence as the delay is reduced and predicts an increased tendency towards intermittency at larger flux Richardson numbers. The results support an interpretation of intermittent turbulence in inclined gravity currents as a delay-induced oscillation arising from the finite adjustment time of turbulence to changes in the mean flow.

We investigate the collision dynamics of hydrodynamically interacting inertialess spherical particle pairs sedimenting in a homogeneous isotropic turbulent flow. The analysis focuses on the rapid-settling limit, in which the particle settling time across a Kolmogorov eddy is much shorter than the Kolmogorov time scale. We also consider continuum breakdown during lubrication interactions, which is important when the separation the particles is comparable to the $O(100)$ nm mean-free path of a gaseous media. Owing to the sub-Kolmogorov particle sizes considered here, we approximate the local flow field in the vicinity of a particle pair as a stochastic linear flow induced by the background turbulence. In the rapid-settling regime, the cumulative effect of turbulent strain fluctuations is weak, and the relative particle motion may therefore be described as a diffusive process. In addition, hydrodynamic interactions generate a net relative drift between the particle pairs. We obtain the hydrodynamic diffusivity and relative drift velocity from the Lagrangian autocorrelation function of the fluid velocity gradient evaluated along the settling trajectory. The rapid-settling assumption further enables us to relate the autocorrelation function to the turbulence energy spectrum. Using these results, we solve the advection-diffusion equation for the pair probability density function to determine the collision rate. We show that the ideal collision rate increases monotonically with increasing relative strength of gravity to turbulence, whereas the collision efficiency decreases monotonically over the same range.

This paper is associated with a video winner of a 2025 American Physical Society's Division of Fluid Dynamics (DFD) Gallery of Fluid Motion Award for work presented at the DFD Gallery of Fluid Motion. The original video is available online at the Gallery of Fluid Motion, https://doi.org/10.1103/APS.DFD.2025.GFM.V045. We investigate bioconvection in a colony of Thiovulum sp. ST bacteria, a recently isolated enrichment culture, confined within a Hele-Shaw chamber. Driven by chemotactic and gravitactic responses, the cells collectively develop striking emergent patterns and convection-like dynamics. Starting from a dense, homogeneous suspension, the swimming bacteria generate large-scale bioconvective flows within minutes. Although these flows resemble thermal convection, they arise in the absence of an imposed temperature gradient. Instead, they arise from the collective swimming of bacteria responding to oxygen gradients and gravity.

This study investigates the spatial development and recovery of near-wall flow steering induced by yawed widely-spaced riblets. To accurately measure the mean spanwise velocity in the near-wall region, we developed a unique, customised rotating probe equipped with a single hot-wire. This novel technique demonstrates high sensitivity, reliably detecting small spanwise velocity components. We applied this probe to map the flow over straight-to-yawed and yawed-to-straight riblet configurations. For the straight-to-yawed transition, the downstream distribution of the mean spanwise velocity closely aligns with the step-forced spatial Stokes layer (SSL) solution, achieving an equivalent active wall motion of $V^+_\mathrm{equ} \approx 2.0$. This offers a simplified analytical pathway for future passive flow manipulation studies. Conversely, flow recovery over the yawed-to-straight configuration diverges from the SSL step-down phase prediction. Instead, a spatial lag was observed where crest-adjacent flow recovered quickly, but significant residual spanwise flow ($V^+ > 0.5$) persisted far downstream ($\widehat{x}{}^{+} > 1000$).

Here, an improved phase-field method for simulating the impact dynamics of solidifying molten metal droplets is developed using targeted free-energy modifications. Conventional Cahn-Hilliard-Navier-Stokes (CHNS) formulations generally do not capture melt retraction over a solidified portion of the droplet, because the newly formed solid region is not treated as an actual internal boundary in a single-order-parameter diffuse-interface model. As a result, the formulation lacks an internal wetting condition or localized wall-energy mechanism capable of driving melt retraction over a solidified splat. To address this limitation, a diffuse-domain wall-energy term is added to the free energy, with a chemical-potential contribution that is active near the diffuse liquid-solidified-material-gas triple-line region, enabling the remaining melt to retract over a solidified splat. In addition, a solidification penalty term is introduced to immobilize the solidified splat formed during impact and suppress unphysical interface motion caused by residual Cahn-Hilliard diffusion inside the frozen region. The proposed formulation is validated against benchmark experiments on impacts of solidifying tin droplets. The results reveal that the localized wall-energy term captures post-maximum-spread melt retraction, while the penalty term effectively arrests motion of the solidified splat. Qualitative and quantitative comparisons with experiments, volume-of-fluid simulations, and standard phase-field predictions demonstrate that the proposed formulation captures post-maximum-spread melt retraction and provides an accurate estimate of the stabilized final splat diameter.

We investigate the start-up flows through ordered porous media (hexagonal close-packed, face-centred cubic and body-centred sphere packs) by means of direct numerical simulations. The flows are initiated from rest and driven by a constant pressure gradient, allowing us to examine the transient development across a wide range of Hagen numbers. Dimensional analysis identifies two relevant time scales: the viscous diffusion time $\tau_\mathrm{visc}$ and the inviscid time $\tau_\mathrm{inv}$. While the small-time behaviour follows the viscous asymptotics of Johnson et al. [J. Fluid. Mech. 176, 379 (1987)], the subsequent emergence of nonlinear effects is universally governed by the inviscid time $\tau_\mathrm{inv}$, rather than by any critical Reynolds number. At the pore scale, the transient evolution is characterised by the growth of thin vorticity layers on the sphere surfaces, their detachment into the pore space around $t \sim \tau_\mathrm{inv}$, and the formation of inertial cores. Despite geometric differences, these processes occur in a remarkably similar sequence across all three packings. Vorticity magnitude exhibits laminar boundary-layer scaling with Hagen number, while in the body-centred cubic sphere pack case a transition towards turbulent-type scaling is observed. These results establish $\tau_\mathrm{inv}$ as a unifying measure for the onset of nonlinearity in strongly accelerated porous media flows, with direct implications for the modelling of unsteady transport in natural and engineered systems.

Aft sections of military cargo aircraft employ flat surfaces at high upsweep angles to accommodate ramp doors, producing flow features that affect cargo-drop accuracy, paratrooper safety, and aerodynamic performance. Fundamental studies have primarily examined near incompressible flow over a canonical surrogate consisting of a freestream aligned cylinder with a planar, sharp edged upswept base. The flow exhibits peripheral separation, a horseshoe vortex, and a counter-rotating streamwise vortex pair that persists downstream. The present investigation delineates the effects of compressibility on the wake and examines how these effects depend on basal upsweep angle. Wall-resolved large-eddy simulations are performed at Mach numbers of $0.1$, $0.3$, and $0.5$ for upsweep angles of $32^\circ$ and $45^\circ$ at a nominal Reynolds number of $25{,}000$. For the $32^\circ$ afterbody, increasing Mach number enlarges the upstream recirculation region and delays vortex-pair formation, while these effects diminish downstream. For the $45^\circ$ afterbody, similar recirculation-region growth triggers a bifurcation at Mach~0.5 from the vortex-pair state to a broad separated turbulent wake. A descending-Mach sequence to 0.3 and 0.1 reveals hysteresis, with the separated-wake state persisting at lower Mach numbers and remaining robust to Reynolds-number variation. Thus, both states can occur at identical Mach and Reynolds numbers, with topology and pressure loading governed by Mach number history.

A major challenge in the field of heterogeneous catalysis is selecting an optimal catalyst support structure. Commercially available structures can be easily manufactured at scale, but their stochastic nature makes their chemical and transport properties suboptimal. This is particularly relevant for gas-generation reactions, where non-uniformity of a porous structure leads to bubble trapping. Such trapping impedes the flow of reactants to catalyst sites, leading to conversion inefficiencies. Previous experimental work demonstrated that spinodally-derived architectures, in particular bicontinuous interfacially jammed emulsion gels (bijels), can alleviate these issues and deliver superior performance. However, to the best of our knowledge, numerical studies to optimize the operating conditions for such a morphology have not been performed yet. In this work, we aim to close this gap using color-gradient lattice Boltzmann simulations of reactive flows with a novel central moments collision operator. We develop an analytical model to predict catalyst performance based on our simulation data. Our findings show that this type of morphology can achieve very high conversion efficiencies. Moreover, we demonstrate that its catalyst performance can be optimized using superhydrophilic surface coatings.

Geological fractures exhibit heterogeneous aperture fields that localize flow along preferential pathways and produce nonuniform fluid-matrix contact times. Heat transport combines channelized advection with conductive matrix exchange. For non-Newtonian fluids, shear thinning biases the aperture-to-flux mapping toward larger apertures, while yield stress suppresses flow below the mobilization threshold. A stochastic semi-analytical model represents aperture classes as parallel pathways with constitutive fluxes and superposes their channel-scale advection-conduction solutions for a semi-infinite matrix. Matrix diffusion sets the late-time scalings of breakthrough curves and front moments, while aperture variability and rheology control their amplitudes and crossover. Rheology affects spreading through high-order flux-weighted aperture moments that determine persistent inter-channel variance. For monomial flux laws, the reduced late-time amplitudes collapse onto a single similarity variable, so shear thinning acts as amplified effective aperture variability. Global sensitivity analysis attributes flux reweighting to shear thinning and hydraulic accessibility to yield stress. Two-dimensional simulations verify the construction for Newtonian and power-law flow. A connected rough-field comparison at low aperture variability recovers mean advance and outlet survival but overestimates variance, so Ellis and Herschel-Bulkley results remain independent-channel predictions. The model provides an interpretable reference limit separating rheology-controlled hydraulic selection from matrix-controlled transport.

Weak shock reflection at nearly glancing incidence is governed, after the transonic weak-shock scaling, by a self-similar unsteady transonic small-disturbance (UTSD) free-boundary problem. We derive and differentiate the first finite-strength perturbation of the corresponding isentropic potential-flow problem along paths of fixed canonical incidence $a=\alpha/\delta$, where $\mu=\delta^2=2(M^2-1)$. A second-order refluxed adaptive finite-volume method, exact discrete tangents and adjoints, and a Rankine–Hugoniot-constrained fitted principal front give the fixed-$a$ canonical sensitivity $H_{2,a}^{\PF}(0.5;1.4)=-0.217\pm0.012$; a fully differentiated physical back-map gives the diagnostic fixed-$a$ coefficient $K_{2,a}^{\PF}\simeq-0.266$. We derive the exact chain rule that converts these quantities to the distinguished fixed-$\lambda$ path $\lambda=(M-1)/\alpha^2$, showing explicitly that the conversion requires the independent incidence derivative of the leading UTSD branch and therefore cannot be inferred from the fixed-$a$ calculation alone. A matched-boundary self-similar Euler study with strength-dependent refinement contains 21 qualified nonlinear states and 252 evaluations of a common front-functional family. Coupled extrapolation gives $g_0^{\Eul}=0.510\pm0.006$ and the physical leading-angle coefficient $G_0^{\Eul}=0.256\pm0.004$, consistent with the shock-fitted UTSD limits. A separate same-strength phase-bracket audit using 18 Euler states shows that the captured-shock subcell phase is comparable to the desired cubic signal. The resulting finite-resolution Euler secants are compatible with the potential-flow correction, but their $\rho=h_\eta/\sqrt\mu\to0$ extrapolation is not model-stable. Thus leading-order Euler–UTSD correspondence is numerically verified, whereas cubic-order Euler correspondence remains unresolved.

We investigate the nonlinear shape dynamics and autonomous propulsion of actively driven quasi-spherical vesicles with locally inextensible membranes at low Reynolds number. Starting from Stokes hydrodynamics, linearized membrane elasticity, and harmonic active forcing, we derive a reduced description in terms of spherical harmonic deformation modes. The global area constraint enforced by local inextensibility is the sole source of dynamic nonlinearity. It confines the dynamics to compact manifolds in the space of possible shapes. Autonomous propulsion arises through nonlinear mode coupling and is determined geometrically by the oriented area swept by the trajectories in shape space. For two active modes, the dynamics reduces to a periodically driven phase equation exhibiting synchronization, phase slips, and mode locking. Introducing a third active mode fundamentally changes the dynamics, giving rise to quasiperiodic invariant tori and resonant periodic cycles. A recurrence diagnostic reveals the resulting resonance structure, while fluctuations of the cycle-averaged propulsion provide an experimentally accessible signature of the underlying shape dynamics. Our results demonstrate that, for actively driven vesicles, a geometric constraint is sufficient to transform an otherwise linear dynamical system into one exhibiting rich nonlinear dynamics.

One of the most critical challenges in turbulent combustion modeling is the chemical source closure. In the recently developed filtered turbulent flame model (FTFM), a oneto-one correspondence between filtered scalar quantities and filtered chemical sources can be constructed by inversely solving the filtered flame equations, without the use of conventional presumed probability density functions (PDFs). However, the turbulence induced flame wrinkling, and thus the enhancement of the chemical source, has not been explicitly considered at the resolved scale. In the present study, the wrinkling effect is analytically quantified in a counterfow flame setup, from which FTFM is then further updated by incorporating such a physics grounded stretching correction on the chemical source. The satisfactory accuracy and robustness of the present model are justified from case tests of the non-premixed Sydney swirl flame and the Delft III flame.

Morphing flexible multibody aircraft require structural strain, aerodynamic geometry, surface velocity, and generalized loading to remain compatible as joints and flexible components change configuration. A compositional formulation is developed around an assumed material attachment between each lifting surface and a geometrically exact beam. Separating the component root pose from the section field shows that the body strain and elastic potential of a component depend on its own elastic coordinates, while upstream motion enters kinetic terms and external-load pullbacks. At element level, an exact relative logarithm $d$ supplies strain and potential energy, whereas a reference-anchored section coordinate $\sigma$ supplies deformed section geometry. Finite-order expansions retain the finite reference geometry exactly and truncate only endpoint perturbations. The attachment map then generates surface points, tangents, normals, velocities, and force Jacobians from common section kinematics. Euler–Poincare beam balance, moving-surface potential-flow relations, graph cotangent assembly, and the associated semidiscrete power identity are stated in a common twist–wrench convention. Collocation, pressure, equivalent-load, and structural-station sites are distinguished to expose their approximation errors. Verification gives the expected $N_d+1$ convergence order for degree-$N_d$ relative-log expansions. In a geometrically nonlinear cantilever comparison, a cubic static-manifold correction reduces mean full-record displacement error from $0.479$ to $0.255$ over four completed load cases. These results provide structural and interface-level evidence rather than validation of a complete aircraft aeroelastic prediction.

The Maxwell-Demon nonequilibrium molecular dynamics method for measuring the shear viscosity divides the fluid simulation cell into two regions of width w in the $x$-direction, with the particles allowed to move freely between the two sides. The Demon maintains equal and opposite regional average particle velocities in the $y$-direction by applying an acceleration needed to sustain the nonequilibrium steady state (NESS), which includes both total force balance and an instantaneous correction for diffusion of particles across boundaries. We show that the driven velocity profile in x is a natural hydrodynamic response and not imposed by the constraint. For the present shear geometry, the measured NESS profile in Eulerian slabs is represented accurately by a piecewise parabolic form, reminiscent of planar Poiseuille flow. The parabolic profile gives an estimate of the kinematic viscosity from work done on the shearing fluid, as well as an entropy production estimate, derived from heat removal by the Nosé-Hoover thermostat that keeps each regional average temperature constant. We show that the mechanical work supplied by the Demon equals the heat removed by the thermostat. Once NESS driving is removed, the parabolic velocity profile relaxes exponentially in short order to sinusoidal, the natural transverse momentum-diffusion eigenmode. Our Maxwell-Demon shear method provides a direct NEMD formalism for obtaining shear viscosity from momentum diffusion, work, and entropy balances. Our results in three dimensions for increasing system size $N$ (the number of particles) demonstrate that shear viscosity approaches an asymptote (the so-called hydrodynamic limit) as $N^{-1/3}$.

Stochastic geometric mechanics (SGM) is known for its potential utility in quantifying uncertainty in global climate modelling of the Earth's ocean and atmosphere while also preserving the fundamental advective transport properties of ideal fluid flow. This paper is a pedagogical review of the recent developments of the mathematical framework of stochastic geometric mechanics obtained from Lie group-invariant stochastic variational principles in the context of model building for upper ocean dynamics, The paper is divided into the following five parts. Part I discusses the origins of geometric mechanics applications in deterministic fluid dynamics. Part II focuses on the example of the deterministic 3D Euler Boussinesq (EB) equations. Part III adds stochastic transport to the 3D Euler Boussinesq (EB) and derives its SALT equations. (SALT is the abbreviation of Stochastic Advection by Lie Transport.) Part IV focuses on Lagrangian Averaged Stochastic Lie Transport, abbreviated as LA-SALT. LA-SALT treats atmospheric `climate' as the ensemble expectation, while the atmospheric `weather' is treated as a field of pathwise fluctuations, as discussed in Ed Lorenz's famous 1995 lecture. Part V applies SALT and LA-SALT to create stochastic Ocean–Atmosphere Models, abbreviated as SOAM.. The SOAM approach brings us back to Hasselmann's 1976 paradigm, which decomposes a general climate model into its deterministic and stochastic parts.

A flexible body placed in a steady flow bends to reduce drag. This self-streamlining is a hallmark of fluid-structure interaction (FSI). Granular-structure interaction is equally ubiquitous in nature. However, it remains poorly understood. Thus, we investigate inertial granular-structure interaction (IGSI). Specifically, ejection induced by a flexible plate sweeping a granular bed is experimentally examined. We find that a faster sweep results in less ejection, particularly for a flexible plate. To understand the underlying physics of this behavior, a dimensionless number Sk is introduced as the ratio of the plate elastic timescale to the sweep timescale. At $\mathrm{Sk} \lesssim 1$, the plate deflection follows the self-streamlining law of FSI and induces substantial ejection. At $\mathrm{Sk} \gtrsim 1$, on the other hand, the plate skims the bed and the mass of ejected grains decreases sharply. Sk organizes IGSI as the granular counterpart of FSI.

We demonstrate optical coherence tomography (OCT) as a measurement technique for dense, optically opaque suspensions. Conventional optical methods cannot access the interior of such suspensions. OCT resolves individual potato-starch particles (${\sim}20~\mathrm{\mu m}$) as distinct scatterers, even though the suspension appears opaque to the eye. By tracking the vertical centroid position of the particle-laden layer $\langle Z \rangle(t)$ and the supernatant boundary $Z_\mathrm{sup}(t)$ in the same OCT image sequence, we obtain the instantaneous settling velocity $V(t)$ and the time-evolving effective volume fraction $\phi_\mathrm{eff}(t)$ simultaneously and continuously in time. To our knowledge, this is the first measurement to combine settling velocity and particle concentration into a single continuous trajectory within one sedimentation run. Conventional batch measurements yield only one velocity value per run. We applied this method to dense potato-starch suspensions, varying the initial volume fraction $\phi_0$ from 0.30 to 0.50 and the solvent density $\rho_\mathrm{L}$ from 1.0 to $1.3{\times}10^3~\mathrm{kg~m^{-3}}$ using aqueous sodium polytungstate solutions. The normalized velocity $V/V_\mathrm{Stokes}$ plotted against $\phi_\mathrm{eff}$ collapses onto a common trend consistent with both the Krieger–Dougherty model and the Richardson–Zaki law over $\phi_\mathrm{eff} \simeq 0.30$–$0.52$, confirming that the method captures physically reasonable hindered-settling behavior. These results establish OCT as a viable tool for probing internal dynamics in dense suspensions that were previously inaccessible to optical measurement.

Rayleigh-Bénard convection of large horizontal extent sometimes self-organizes into domain-filling structures. In two dimensions, domain-filling rolls persist - from some but not all initial conditions - when velocity boundary conditions are stress-free. When velocity boundary conditions are no-slip, domain-filling rolls are not found with fixed-temperature thermal boundary conditions, but they have not been sought with fixed-flux thermal boundary conditions. Here we explore the latter missing case, which is hard to predict because no-slip boundaries do not encourage domain-filling rolls, but fixed-flux boundaries give domain-filling structures in three dimensions for either boundary condition on velocity. We simulate convection with fixed-flux, no-slip boundaries in two-dimensional domains with horizontal period 20 times their height and at various combinations of the fixed-flux Rayleigh number R and Prandtl number Pr. Domain-filling rolls, which are easily found when they are weakly nonlinear at small R, are continued to other (R,Pr) by changing these parameters slowly in time. The (R,Pr) regime where we find domain-filling rolls is substantial but has a boundary. When R is too large or Pr too small, relative to each other, a domain-filling roll pair breaks up into two pairs. These findings contrast with other combinations of dimension and boundary conditions, where scale selection has not been seen to depend strongly on parameter values. The present case helps disentangle competing effects of dimension and boundary conditions, and it offers a more stringent test for explanations of scale selection that have been proposed.

Buoyancy points straight upward, yet in the absence of external forces, a rising air bubble may trace a spiral through the water. This phenomenon was already observed by Leonardo da Vinci. We show that the gravity-free, three-dimensional two-phase incompressible Euler equations with surface tension already admit such a motion. For every nonnegative inner-to-outer density ratio and every sufficiently small positive axial Weber number, we construct smooth near-spherical bubbles and drops with compact interfaces and phasewise irrotational flow. They are stationary in a frame translating along and rotating about a fixed axis, while their centroids lie off that axis and therefore trace genuine circular helices in laboratory coordinates. These helical relative equilibria form a symmetry-breaking branch that issues from the axisymmetric family of rectilinearly translating solutions and is generated by the spherical harmonic mode of degree two and azimuthal number one. The mathematical construction exploits an overdetermined free-boundary formulation, its variational structure, and an $\mathrm{SO}(2)$-equivariant Lyapunov-Schmidt reduction. Particular difficulties in the form of resonances arise at a discrete set of density ratios, including the frequently studied vacuum case with vanishing inner density. Complementary to our existence result, in the absence of surface tension and if the inner density is not larger than the outer density we prove that the model admits neither nontrivial translating relative equilibria nor relative equilibria with helical centroid trajectories. As a byproduct, we show that every regular finite-energy stationary vortex sheet of spherical topology for the homogeneous three-dimensional Euler equations, with irrotational flow on both sides, is trivial.