Papers

Droplet-laden cloud dynamics emerge from coupled phase transitions and turbulent mixing. Combining experiments and mean-field theory, we establish scaling laws for steam-injected clouds in unsaturated air seeded with condensation nuclei. Depending on whether the cloud is confined to the initial momentum-dominated zone of the jet or extends into the buoyancy-dominated plume, two asymptotic regimes arise. In the momentum regime, cloud dimensions scale with the injector radius. In the buoyancy regime, size follows a \(\dot{m}^{2/5}\) scaling, where \(\dot{m}\) is the injected steam mass flow rate. Assuming inter-droplet air remains humidity-saturated, our mean-field analysis predicts cloud persistence governed by enthalpy and humidity mixing. These findings bridge microphysical processes and macroscale dynamics, offering a quantitative framework for atmospheric cloud evolution.

A recent study of various turbulent flows showed that intermittency and dissipation exhibit consistent scaling behavior beyond homogeneous isotropic turbulence (HIT). For turbulent wakes, grid turbulence, and a turbulent jet, the dissipation parameter $C_\varepsilon$ and intermittency parameter $\mu$ vary while their product remains constant, independently of the Reynolds number $Re_\lambda$. Here, we extend this result from centerline hot-wire measurements to non-centerline data, including regions with significant shear. We identify two additional relations: $(\gamma-1)C_\varepsilon$ remains constant, where $\gamma$ is the absolute inertial-range spectral slope, and $\gamma$ varies linearly with the Kolmogorov parameter $C_k$. The main constraint is that the large-scale increment probability density function is Gaussian, indicating random large-scale driving. These relations reduce the varying quantities $C_\varepsilon$, $\mu$, $\gamma$, and $C_k$ to three constants and define a turbulence state that can be characterized by any one of these parameters, independently of shear, flow type, or Reynolds number. We further show that, for a given flow configuration, $C_\varepsilon$ depends on $Re_\lambda$ and turbulence intensity $TI$. Because the four parameters vary over broad ranges around their nominal HIT values, we interpret this framework as a generalization of HIT to inhomogeneous turbulence. Extensive robustness tests show that the results are insensitive to the specific post-processing method. We also analyze their streamwise and spatial evolution and show that a measured $C_\varepsilon$ can be used to predict $\mu$, $\gamma$, and $C_k$ within quantified uncertainty. The framework thus links key leading-order features of turbulence while recovering established HIT trends.

One of the main challenges limiting the maximum forward speed of high-speed rotorcraft is the development of reverse flow on retreating blades. Flow separation at the sharp aerodynamic leading edge during reverse flow leads to negative lift, high drag, high pitching moment, and a pitching moment impulse. The study experimentally investigates the effectiveness of a sinusoidal trailing edge as a passive method for the control of reverse flow, under static pitching conditions. Three NACA 0015 two-dimensional blades were evaluated under both reverse and forward flow conditions. A baseline blade with a straight trailing edge and two modified blades with sinusoidal trailing edges, each having a spatial amplitude of 10% of the chord and wavelengths of 5% and 10% of the chord, respectively. Experiments were conducted at chord-based Reynolds numbers of 70000 and 140000, consisting of planar Particle Image Velocimetry and aerodynamic load measurements. The study shows that a sinusoidal trailing edge significantly mitigates flow separation, leading to a large reduction in the associated problems over a wide range of angles of attack. The problem of pitching moment impulse is eliminated. The sinusoidal geometry with a lower wavelength of 5% of the chord shows comparatively better performance. Additionally, both sinusoidal geometries minimally affect the forward flow characteristics.

We observe the emergence of extreme events in an active fluid involving two distinct chemical species that regulate active stress. One species is slow diffusing and the other is fast diffusing, and the growth of the fast-diffusing species is modelled using a nonlinear logistic term. We demonstrate the presence of extreme events in the temporal evolution of the concentrations, as well as in the spatial profile of the system,in regimes of merging-emerging soliton-like dynamics and spatio-temporal chaos, through analysis of the time-series, bifurcation diagrams, probability distribution functions of the concentration of the two species, return maps and distributions of inter-event intervals. Interestingly, we also find evidence of pronounced bunching of extreme events and super-extreme events in the slow chemical species in the soliton-like regime. We go on to systematically explore the dependence of the extreme event occurrences on the Péclet number and the strength of the nonlinear growth term, and find that the probability of extreme events increases after a critical Péclet number, while increasing the nonlinearity suppresses extreme events. Lastly, in order to gain further insight, we investigate a modified mode-truncated reduced order model comprising of coupled differential equations mimicking this active fluid system. We find that this reduced order model also exhibits extreme events whose emergence is correlated with a sudden expansion in attractor size due to a crisis arising from attractor collision.So these results demonstrate the existence of extreme events in an active fluid system, and are of potential relevance to biological phenomena where active transport plays an important role.

We introduce a compatibility–realization framework for constructing singular solutions to the forced three-dimensional incompressible Navier–Stokes equations. The framework separates exact balance and matching constraints, expressed in a chosen geometric representation, from the sufficient conditions of a particular completion scheme. Completed realizations are organized by observable fibers, distinguishing prescribed core kinematics from the surrounding force and transport mechanisms. Conditional on the source-dependent completion input specified in this paper, we obtain families with zero initial velocity, smooth compactly supported forcing, bounded kinetic energy and finite-time unbounded velocity. These include a reversal of leading viscous work at fixed axis motion and first velocity gradient, a signed transition through a stationary singular center under one fixed canonical pressure trace, and a connected stretching family crossing the threshold of zero leading direct viscous vorticity supply. Near a critical profile, the realizable trace data contain a neighborhood in an infinite-dimensional space of analytic axial-velocity and pressure perturbations, while the central first velocity gradient and the critical vorticity balance remain fixed. We also identify the full kernel and an explicit right inverse of the joint angular-transport operator in the chosen realization class. The results establish non-rigidity within observable fibers and show that axis kinematics alone do not determine the local pressure–viscosity partition. Direct profile selection and dynamical stress generation are treated as coupled components of construction, rather than as definitions of the entire design space.

We derive computable upper bounds on the maximum wave speed in the Riemann problem for the compressible Euler equations with a composite equation of state, in which the pressure is the sum of a convex hydrodynamical part and a barotropic correction. The correction may destroy the convexity of the equation of state, so that the Riemann solution can contain composite shock-rarefaction waves. The bounds are obtained by comparison with the auxiliary Riemann problem for the hydrodynamical equation of state alone. The bounds hold irrespective of whether the outer waves are elementary or composite and require no case distinction on the wave type.

Biological swimmers and flyers exploit unsteady vortices for propulsion, whereas engineered vehicles usually suppress them as disturbances. Learning such flow exploitation in machines is difficult because real-fluid interaction data are scarce and unstructured exploration is unstable in high-dimensional, history-dependent flows. Here we present REEF, a co-designed physical-learning framework that integrates SHOAL, an eight-channel high-throughput array for real fluid–structure interaction, with V-STAR, a staged algorithm that converts these interactions into policies through imitation, offline internalization, and online adaptation. Across lift-based, drag-based, and momentum-jet propulsors, REEF expands the attainable force envelope to more than twice that of parameterized search. Particle image velocimetry shows that these gains arise from coordinated vortex formation, growth, and force projection, rather than refinement of a fixed motion-to-force mapping. Force-trained policies transfer zero-shot to free-moving robots whose body motion changes the surrounding flow, suggesting that REEF learns transferable wake-coupling principles for embodied propulsion in unsteady fluids.

Shock fronts and vortex cores often coexist and overlap in compressible flows, where walls, wakes and shear layers also produce strong gradients. Refining a network to detect one structure can therefore degrade its prediction of the other without revealing the loss in the optimized score. We present a task-preserving formulation for simultaneous shock and vortex-core segmentation. The experiments use two solvers and include supersonic diamond-airfoil flows at several incidences and Reynolds numbers, together with circular- and elliptical-cylinder flows. A shared primitive encoder feeds separate shock and vortex decoders with independent sigmoid outputs, allowing the two classes to overlap. Adaptation is confined to the relevant branch: zero-initialized adapters supply rotational diagnostics only to the vortex decoder, while corrected shock supervision updates only the shock decoder. All dependencies of the protected output remain fixed, and bitwise equality is verified on every evaluation field. Compression and conservation-jump (Rankine–Hugoniot) signatures provide weak shock supervision; rotation and topology provide vortex candidates. Analytical oblique-shock rays, Billig's bow-shock correlation and an isentropic vortex supply references independent of these labels. Under the same corrected-target budget, the restricted model and a capacity-matched shared-decoder U-Net obtain comparable shock agreement. Their airfoil vortex-core Dice overlap scores, however, are 0.83 and 0.08, respectively. A soft retention penalty recovers most of the U-Net's lost core agreement. The frozen shock-adapted models locate the tested analytical oblique-shock rays within 0.003 chord. An additional branch identifies expanding-flow regions, which are distinguished from centred Prandtl–Meyer expansion fans through comparison with ideal shock–expansion theory.

Hypothesis: The presence of a polymer in a droplet plays a significant role in contact line dynamics following droplet impact onto a solid substrate. We hypothesize that the polymer concentration modifies the post-impact wetting behavior. Experiments: In our experiments, we visualize the contact line dynamics of polymer-laden droplets impacting onto a flat sapphire substrate. Using high-speed side and bottom-view imaging, we analyze droplet spreading and retraction over a broad range of Weber numbers (10-700) and polyethylene oxide (PEO) concentrations (0-400 ppm). To systematically characterize the contact line dynamics, we used image processing to quantify the maximum spreading factor, receding velocity, and receding contact angle. Findings: The effect of the polymer on the spreading phase is negligible, as the maximum spreading factor follows classical inertial-capillary scaling, showing that polymer addition produces no measurable change in the maximum spreading under the present conditions. In contrast, the receding phase exhibits a strong dependence on the polymer concentration: increasing the PEO content leads to a significant reduction in the receding velocity and dynamic contact angle, while the spreading dynamics remain water-like. These results are consistent with an additional resistance associated with the receding contact-line region, while the microscopic origin of this resistance remains unresolved. To capture this behavior, we propose a scaling model that relates the normalized retraction velocity to the receding contact angle and impact conditions. The model effectively collapses the experimental data across all investigated concentrations and Weber numbers.

The present work proposes a novel Lagrangian dynamical theory for the velocity gradient tensor (VGT) $\bm{A}\equiv\bm{\nabla}\bm{u}$, formulated on the basis of its canonical real Schur form.... The proposed theory is expected to be useful for understanding the evolution dynamics of the VGT rotational invariants and intrinsic vorticity modes in a variety of vortical flows.

Thixotropic and viscoelastic materials both retain their deformation history and are difficult to distinguish. However, a stationary linear response obeys the Kramers-Kronig relations. A structure that evolves during a sweep renders the spectrum inconsistent with them, so the residual from one chirp and no model measures nonstationarity. It survives the small-amplitude limit, returns to its baseline at any stationary state, and, for bounded restructuring, peaks near the structural time. We verify this on eight models and demonstrate it on published spectra of a thixotropic silica suspension.

The present study analyzes encounters of a vertical vortex with a tailless NACA 0012 delta wing with a sweep angle of 60 degrees at an angle of attack of 10 degrees and Reynolds number 1000 using direct numerical simulations. Motivated by the need to fly in complex settings such as urban canyons, mountainous areas, wildfires, or vehicle swarms, we focus on the influence of a vertical vortex gust on the delta wing, complementing previous efforts that predominantly studied spanwise and streamwise vortex encounters. The vertical orientation can impose strong transient loads on all six degrees of freedom, and particular attention is devoted to connecting these loads to the flow response. By varying the gust's lateral position, size, and strength, we find a fundamental process that characterizes vertical vortex gust encounters. The gust consistently imprints a low-pressure core on the wing surface. The evolving core creates an adverse pressure gradient that distorts the baseline skin-friction pattern and induces separation that lifts the boundary layers. These lifted boundary layers are elongated and twisted by the gust's strain into lobed vortical structures, which are shown by force element analysis to make a critical contribution to the lift. As the gust's lateral position is varied from root to tip, the low-pressure core and the induced flow separation are increasingly localized around the leading edge closest to the gust, modifying the lobe evolution and leading to complex dependence of the force and moment response on gust position compared to the largely monotonic dependence on gust size and strength.

Accurate estimation of unsteady aerodynamic flows from sparse measurements remains a fundamental challenge, particularly in the presence of strong disturbances, unknown body kinematics, and incomplete observations. This study presents a data-driven sequential estimation framework for joint reconstruction of unsteady flow fields, aerodynamic loads, and airfoil kinematics from sparse measurements. The approach combines a kinematics-aware nonlinear flow autoencoder with online filtering of streaming measurements. In the resulting reduced-order representation, the forecast and observation operators are learned directly from data, with each forecast-assimilation cycle requiring only a few milliseconds. The framework is evaluated on two-dimensional incompressible flow over an airfoil subjected to random vortical gusts while undergoing arbitrary pitch-up motions. Results demonstrate accurate reconstruction from surface pressure sensors, vertical lines of vorticity sensors, and synthetic velocimetry data with representative shadow regions. The transient informativeness of pressure sensors is quantified through their time-varying contributions to dominant observation modes. While lift is reconstructed from the leading modes, drag, pitch angle, and angular velocity require higher modes. Incorporating limited off-body measurements alongside surface pressure improves observability and reduces estimation error and uncertainty.

Can attention move a shock across several cells in one update without breaking it? We develop a conservative, fixed grid finite-volume scheme in which a CFL-conditioned attention flux selects upstream information according to the transport required by the current time step. One-dimensional inviscid Burgers transport is used as the central mechanism test: the same learned flux remains reliable in the conventional small-step regime and, with a time step four times larger, preserves sharp shocks while using one stage per update. A standard fifth-order WENO scheme with third-order strong-stability-preserving Runge-Kutta time integration (WENO-5+SSP-RK3) is included alongside controlled Forward Euler comparisons to separate flux selection from time integration. The learned attention shifts upstream with the local transport reach and becomes more selective near shocks; inference-time interventions and retrained ablations show that transport-scale information and state-dependent selection contribute directly to performance. Directional two-dimensional scalar Burgers transport and the one-dimensional shallow-water system then test whether the conservation-scale-selection principle transfers beyond the original scalar setting. The results support attention as a learnable information stencil for conservative large-step shock transport, while identifying finite candidate reach and problem-dependent robustness as the present limits.

High-order entropy stable (ES) methods have attracted much interest in the computational fluid dynamics (CFD) research community over the past decade, as we seek to simulate practical engineering applications with high accuracy and robustness. As industry delves further into supersonic flight and space travel, we need schemes capable of modelling flows with multiple chemical species that interact and mix within the fluid. As such, there is a considerable amount of effort currently being focused on developing high-order ES schemes for multi-species (MS) flow. A key ingredient for these entropy stable schemes is the two-point flux, which determines the properties of the resulting high-order discretization. As such, the numerical scheme can possess important properties such as entropy conservation and kinetic energy preservation only if the two-point flux also mirrors these properties. At present, in the literature, the two-point fluxes used for high-order ES multi-species solvers are not kinetic energy preserving (KEP). In this short communication we develop a two-point flux that is both entropy conserving (EC) and kinetic energy preserving by adapting Ranocha's single-species methodology for multi-species. The newly proposed EC and KEP flux is tested against existing two-point fluxes in the literature using the density pulse advection and Taylor-Green vortex cases. It is shown that EC and KEP flux retains both properties while the other fluxes only retain one of the desired properties. Since the two-point flux dictates the properties of the resulting scheme, the proposed flux is the ideal option for simulating multi-species compressible flow.

We present a Godunov-type mesh-free hydrodynamics scheme for the numerical simulation of weakly compressible single- and multiphase flows. The method employs a consistent second-order spatial reconstruction of the primitive variables at the inter-particle interface, applies slope limiting, and solves the moving Riemann problem using a subsonic HLLC approximate Riemann solver. Particles are advected with a quasi-Lagrangian material velocity incorporating an inherent self-relaxation mechanism that suppresses the formation of anisotropic particle distributions. Near material interfaces, one-sided renormalized gradients are used to handle density discontinuities and slope limiting is applied for each phase independently. The Lagrangian character of material interfaces is preserved by constraining the relaxation mechanism to act tangentially to the interface. For same-phase particle interactions, the Riemann problem is centered at the arithmetic mean of the two material velocities. For cross-phase interactions, it is centered at the contact-wave velocity. This choice naturally eliminates spurious mass fluxes across phase boundaries. The methodology is validated on a set of challenging benchmarks, including the inviscid Taylor-Green vortex, material interface advection, surface gravity waves near hydrostatic equilibrium, the Kelvin-Helmholtz instability, and the Rayleigh-Taylor instability under gravitational stratification.

Laser Powder-blown Directed Energy Deposition (LP-DED) offers flexibility for process and materials and high productivity (~5 kg/h), but process-induced pores often compromise mechanical properties. This study utilizes in-situ X-ray synchrotron imaging to compare pore formation mechanisms in Ti-6Al-4V (Ti64) and stainless steel 316L (SS316L), focusing on the interplay between particle dynamics and thermophysical properties. Four distinct pore formation mechanisms were identified, with most large pores originating from the closure of cavities formed behind incident particles impinging on the melt pool. High Weber number (We >> 1) governs this behavior, indicating that particle inertia, rather than thermophysical property differences, is the primary driver of large pore formation. The study demonstrates that increased energy density leads to larger melt pool volumes, facilitating deeper particle penetration. This greater penetration depth directly correlates with increased pore diameters. While thermophysical properties secondarily influence pore-formation frequency and cavity symmetry, particle inertia remains the dominant factor. These findings provide a physically grounded basis for understanding and controlling porosity in powder-blown DED processes.

For a model of a thin spherical layer of an incompressible rotating fluid subjected to the action of Coriolis and buoyancy forces, both analytical solutions in the form of stationary thermal spots moving along the equator and self-similar solutions are found. The influence of the parameters determining the shape of the spots on their propagation speed and direction is studied. It is found that the speed and direction of motion of the spots depend not only on the sign of their thermal contrast with the background flow, but also on the ratio of the semi-axes determining their shape. In particular, if the ratio of the equatorial semi-axis to the meridional one $b/a$ is equal to $\pi/2$, the spots are at rest. If the spots are elongated along the equatorial axis to such an extent that $b/a>\pi/2$, then “cold” spots move westward and “hot” spots move eastward. Under the condition $0.747\leq b/a<\pi/2$, the situation is reversed: “hot” spots move westward and “cold” spots move eastward. Self-similar solutions in the model are realized strictly in the form of circular thermal spots, and their radius varies according to a power law with a scaling exponent $k$, which is determined by the behavior of the cross-frontal buoyancy gradient on the spot contour. We note that the regime $k=1/4$ is realized under the assumption of a constant buoyancy gradient, while the regime $k=1/6$ corresponds to the conservation of total buoyancy.

Understanding interparticle radiation forces between spherical and nonspherical particles is crucial for the trapping of micro- and nano-objects in acoustophoresis. Most theoretical and numerical models treat spherical particles as identical to estimate interparticle forces, owing to their symmetry. Here, we elucidated interparticle radiation forces between three particles with different shapes and material contrast using experimentally validated numerical models. We performed a frequency sweep of the full acoustic device, accounting for viscous effects in a two-dimensional model, to predict the system's resonance. The pressure field obtained at the resonant frequency was then fitted to a 1D sinusoidal wave and used in a three-dimensional model based on the perturbation technique and tensor integral method to calculate the interparticle forces on the particles due to scattering and re-scattering of the acoustic field. Our results revealed a shape-dependent reversal of interparticle forces, even for positive-contrast materials. We found that for spherical particles, the interparticle forces are independent of material properties when the particles are positioned on the nodal plane. However, material contrast and particle asymmetry significantly affect interparticle forces when particles are placed outside the nodal plane. Finally, we studied the effects of the orientation of the asymmetric particle placed outside the nodal plane on the interparticle radiation forces. Interestingly, for the orientation angle of 0 degrees, the interparticle force is maximum for the asymmetrical particle and then gradually decreases to zero as we increase the orientation angle to 90 degrees. The detailed analysis presented in the paper will facilitate a better understanding of shape-based acoustic manipulation of microparticles.

We describe how strong vorticity fluctuations appear in three-dimensional incompressible turbulence driven by a large-scale random force. By combining analytical and numerical methods, we construct a vorticity instanton: the most probable force history driving the flow to a prescribed large vorticity at a given point and time. We show how the smooth, large-scale force produces a long vortex filament with a thin, viscous-scale core. Our main finding is that the standard mechanism of amplifying a vortex tube through radial compression and axial stretching is only effective up to a threshold value. Producing vorticity above the threshold requires an additional ingredient: vorticity waves propagating along the tube. The final vorticity is the result of two colliding waves. This mechanism has striking similarities to the one recently suggested by OpenAI for creating infinite vorticity in finite-time Navier–Stokes blow-up.

We present a novel, spatially two-dimensional (2D) set of equations to study the propagation of linear deep-water surface waves over a general steady three-dimensional (3D) background free surface flow. No assumptions of a flat background free surface, nor of an irrotational background flow, are made. The resulting model is not only a significant theoretical simplification, but also results in orders of magnitude faster computations. The linearized Euler equations are evaluated on the free surface of the base flow, and a closure condition is proposed to account for the vertical derivatives at the free surface. This generalizes a recent result for purely rotating background flows (Zuccoli, Brambley & Barkley, 2025, arXiv:2405.12078). The final model consists of five coupled first order partial differential equations (PDEs) to be solved on the 2D free surface, involving five unknowns: the horizontal and vertical perturbation velocities; the free surface perturbation height; and unexpectedly the gradient of perturbation pressure with depth at the surface. Two test problems are used to validate the model: a uni-directional vortical depth-varying base flow with a flat free surface; and perturbations to a travelling Gerstner wave solution. Eigenvalues and eigenfunctions of the linearized Euler equations are computed and compared with those of the model. Results show remarkable agreement between the two. Our study finds for the first time, to the best of our knowledge, that sufficiently steep two-dimensional Gerstner waves are unstable.

We present a variational formulation of the induction equation in magnetohydrodynamics (MHD) based on Gauss's principle of least constraint. The central result is the Euler–Lagrange equation $\mathbf{Z}_B = -\nabla h_m$, where $h_m = \mathbf{A}\cdot\mathbf{B}$ is the magnetic helicity density. This reveals that the magnetic helicity gradient $\nabla h_m$ acts as the constraint force maintaining the solenoidality of the magnetic field, exactly as the pressure gradient $\nabla p$ maintains incompressibility in Taha et al.'s pressure-gradient minimization principle. The magnetic helicity density—which, though gauge-dependent locally, yields a gauge-invariant variational principle—naturally emerges as the Lagrange multiplier enforcing $\nabla\cdot\mathbf{B}=0$. At the solution, the field minimizes the norm of the magnetic helicity gradient $\|\nabla h_m\|^2$. This establishes a structural analogy: magnetic helicity is to the magnetic field as pressure is to velocity. The variational principle connects to Woltjer's theorem, Taylor relaxation, and the Hamiltonian structure of MHD.

To elucidate the physical mechanism of turbulence attenuation by polymers at each scale, we conduct direct numerical simulations of homogeneous isotropic turbulence in dilute polymer solutions. We model polymers as FENE dumbbells and simulate them using Brownian dynamics. By visualising the hierarchical structures of coherent vortices, we demonstrate that as the Weissenberg number increases, polymers progressively suppress vortices from smaller to larger scales, attenuating their turbulent energy. While inspired by Lumley's theory, we propose a novel timescale-based framework for analysing this attenuation process using the scale decomposition. We define a scale-dependent Weissenberg number, $\mathrm{Wi}_{\mathrm{sd}}(k)$, as the ratio of the polymer relaxation time to the turnover time of multiscale vortices at each wave-number. We reveal that when expressed in terms of $\mathrm{Wi}_{\mathrm{sd}}(k)$, the energy attenuation rate at each scale collapses onto a single curve that rises at $\mathrm{Wi}_{\mathrm{sd}}(k) \gtrsim 1$, proving that $\mathrm{Wi}_{\mathrm{sd}}(k)$ successfully describes both the onset and the degree of turbulence attenuation at any given scale $k^{-1}$. Furthermore, the scale decomposition uncovers that polymers preferentially align with the turbulent stretching direction at the scale satisfying $\mathrm{Wi}_{\mathrm{sd}}(k) \approx 1$. Based on these results, we establish a physical picture of the polymer–turbulence interaction and explicitly link it to the statistics in turbulence attenuated by polymers.

We investigate pressure imposed two-phase flows in porous media where a Newtonian fluid displaces an immiscible Bingham yield-stress fluid. Using a pore-network model, we identify invasion patterns governed by three dimensionless parameters: the pressure-based capillary number $Ca_P$, the viscosity ratio $M$ , and a number $R_\tau$, the ratio of yield stress to capillary forces. Beyond the classical Lenormand regimes (viscous fingering, capillary fingering, stable displacement), we discover two new regimes: a tree-like pattern at low viscosity ratios and a progressively widening column-like pattern at high viscosity ratios. We characterize all regime transitions and establish a breakthrough condition for the invading fluid.

This study investigates fully developed stratified gas/liquid magnetohydrodynamic (MHD) flow of an electrically conducting liquid and a nonconducting gas in inclined rectangular ducts subjected to a vertical magnetic field. Analytical and numerical solutions for the velocity and induced magnetic fields are obtained in terms of the governing dimensionless parameters for concurrent upward, concurrent downward, and countercurrent flows. Unlike single-phase MHD flow, duct inclination strongly affects two-phase flow by altering the liquid holdup and the relative contributions of gravitational, frictional, and electromagnetic forces. The results reveal a complex interplay among gravity, Lorentz forces, and wall and interfacial shear stresses. These interactions govern the liquid holdup, pressure gradient, multiple steady solutions, flooding limits, local backflow, jet-like velocity structures, and pumping requirements. Wall conductivity critically affects the induced magnetic field and Lorentz force distribution and therefore cannot be neglected, even at very small magnetic Reynolds numbers. Fully insulating ducts generally exhibit the weakest electromagnetic effects and behavior closest to non-MHD flow. Configurations with a conducting bottom wall exhibit substantially stronger electromagnetic effects and greater sensitivity to side-wall conductivity, leading to pronounced changes in the velocity field, liquid holdup, pressure gradient, gas-lubrication effect, and overall pumping-power requirements.

Reciprocal flows repeatedly return fluid elements to their initial positions, producing no net advective transport on time average. Nevertheless, their interplay with diffusion gives rise to nontrivial transport. To describe this phenomenon, we develop a general theory of effective diffusion under two-dimensional linear flows with arbitrary time dependence. By analyzing the advection-diffusion equation, we derive exact expressions for the mean square displacement and the effective diffusion coefficients for extensional, simple shear, and rotational flows within a single framework. We show that the reciprocal flows universally induce diffusion enhancement. The diffusion tensor exhibits pronounced anisotropy, which can result in directional diffusion suppression in spite of the direction-averaged diffusion enhancement. Our results provide a general framework for diffusion control by time-dependent flows and can provide new strategies for transport manipulation in microfluidic and biological systems.

Townsend's attached-eddy hypothesis explains the asymptotic statistics of wall turbulence through a superposition of self-similar attached eddies. At a given observation location, active portions contribute to momentum transfer, whereas inactive portions retain only wall-parallel fluctuations. Since this distinction depends on the observation height relative to the eddy height $y_l$, we extract single-eddy intensity functions from DNS of turbulent pipe flow at $Re_\tau\simeq930$–$6000$. Spectral linear stochastic estimation and differencing isolate wall-coherent contributions at prescribed $y_l$, yielding the streamwise and wall-normal intensity functions $I_{uu}$ and $I_{vv}$. When expressed in $y^*=y/y_l$, both collapse across $y_l$ and share a peak at $y_a^*$, close to the von K'arm'an constant. Near $y_a^*$, both components remain significant, whereas for $y^*\ll y_a^*$, $I_{vv}$ diminishes while $I_{uu}$ remains finite, identifying active and inactive portions. Over the self-similar range, both peak intensities scale as $y_l^{-1}$, and $I_{uu}$ retains this scaling in both portions. Integrating the active portion yields an approximately constant contribution, while the inactive portion recovers the classical near-wall logarithmic variation. An outer logarithmic tendency with a slope close to the Townsend–Perry constant arises from integrating active contributions up to their outer cutoff. At the near-wall peak of $\langle uu\rangle^+$, the scaling steepens to $I_{uu}\sim y_l^{-(1+\beta)}$ with $\beta\simeq1/4$, yielding a finite asymptote with a $Re_\tau^{-1/4}$ defect. The additional exponent is attributed to viscous attenuation of inactive footprints associated with a $y_l$-dependent dissipative scale. Wall-coherent motions also produce a broad outer shoulder, while a localized wall-incoherent residual may contribute to a secondary peak.

The rapid development of liquid-cooled data centers has imposed imperative demands on the performance of liquid cooling plate. The density-based topology optimization (TO) is an effective approach to resolving the growing thermal-hydraulic performance requirements of liquid cooling plate. However, existing TO methods can hardly optimize convective heat transfer directly which is the intrinsic heat transfer mechanism, due to the highly complex and evolving structural topologies, varying flow and temperature fields, making it extremely challenging to explicitly describe the heat transfer coefficient and heat transfer area during TO process. A convective heat transfer topology optimization (CTO) method is proposed in this study, where the iteratively evolving heat transfer coefficient is explicitly depicted by the field synergy theory in the thermal objective, and directly described by the velocity and temperature fields without relying on specific geometry. Combined with the explicit depiction of heat transfer area by the fractal geometry theory, a CTO framework is built for a direct optimization of convective heat transfer under both the laminar and turbulent flow conditions. The CTO tends to generate more hierarchical and directional structural topologies in optimization results, which is conducive to reducing low-velocity stagnation zones and improving flow direction in branched channels, achieving enhanced synergy and thermal-hydraulic performance in the optimized liquid cooling plates. Compared with the TO results without incorporation of field synergy theory, the CTO can reduce average temperature rise by 20% while improving the Nusselt number by 15% under laminar flow conditions, and reduce maximum temperature rise by 10.2% and pressure drop by 25% under turbulent flow conditions.

In this study, a physics-enhanced neural operator framework is proposed to enhance the generalization prediction ability of the cooling layout of a turbine endwall with variable number of film holes. Specifically, inspired by the film cooling superposition principle, we propose a film cooling prediction model, namely superposition-based deep neural operator (SDNO), that divides the endwall temperature field prediction into two stages. In the first stage, the cooling layout of a turbine endwall is divided into several sub-parts with randomly assigned film holes, and a Transformer-based neural operator network, namely Calculate Net, is designed to predict the temperature field of each sub-part. Then, in the second stage, another neural operator network, i.e., Super Net, is trained to combine the temperature fields predicted by Calculate Net for each sub-part and obtain the superposed temperature field of the full cooling layout. Additionally, instead of directly taking the film cooling contours as pixel plots, a signed distance function (SDF) which is sensitive to the variable locations of cooling holes, is designed to encode the location information of cooling holes. Furthermore, the proposed endwall film cooling prediction model is trained with the samples that changing the number of film holes from 1-5 with variable locations. Then, the trained prediction shows excellent generalization prediction ability, which can accurately predict the film effectiveness of the cooling layout with 10-20 film cooling holes that are unseen in the training samples. The proposed SDNO also improves prediction accuracy relative to the fully supervised baseline. With the above, the effectiveness of our proposed prediction model has been well demonstrated.

We survey a geometric approach to the Navier-Stokes equations developed by the author over the past fifteen years. The central object is a Riemannian space of fourteen dimensions whose metric is Ricci-flat precisely on the solutions of the Navier-Stokes system, so that the behaviour of a viscous incompressible fluid becomes a property of curvature. The space decomposes into a flat six-dimensional part and two dual quadruples of coordinates, Eulerian and Lagrangian, which exhibits the two classical descriptions of a fluid as two halves of a single geometry. We also recall the associated six-dimensional metric, whose integrability condition is the incompressibility of the fluid; the link with the projective geometry of second-order ordinary differential equations in the sense of E. Cartan; and the use of Cartan's invariants to construct a three-dimensional analogue of the Taylor-Green vortex. An application of the same construction to the equations of rotation of a rigid body is indicated.