Natural language is a complex system that exhibits robust statistical regularities. Here, we represent text as a trajectory in a high-dimensional embedding space generated by transformer-based language models, and quantify scale-dependent fluctuations along the token sequence using an embedding-step signal. Across multiple languages and corpora, the resulting power spectrum exhibits a robust power law with an exponent close to 5/3 over an extended frequency range. This scaling is observed consistently in contextual embeddings from both human-written and AI-generated text, but is absent in static word embeddings and is disrupted by randomization of token order. These results show that the observed scaling reflects multiscale, context-dependent organization rather than lexical statistics alone. By analogy with the Kolmogorov spectrum in turbulence, our findings suggest that semantic information is integrated in a scale-free, self-similar manner across linguistic scales, and provide a quantitative, model-agnostic benchmark for studying complex structure in language representations.

The law of the wall (LoW) provides a universal description of the mean velocity profile in wall-bounded flows at sufficiently high Reynolds numbers. However, deviations from the LoW are observed at low Reynolds numbers, often vaguely attributed to the so-called low-Reynolds-number effect. These deviations pose significant challenges for near-wall turbulence modeling, as real-world engineering flows span both low and high Reynolds number regimes. Moreover, the absence of a universal scaling for the mean flow underscores distinct flow physics between these regimes. This study introduces a velocity transformation that aligns velocity profiles at both low and high Reynolds numbers with the LoW, via involving the effect of the variation of the total shear stress. The proposed transformation is validated against a wide range of wall-bounded flows, including channel, pipe, Couette, Poiseuille-Couette, and zero-pressure-gradient boundary layer flows, with the Reynolds number ranges from 100 to 10000. Furthermore, a new near-wall model is developed based on this mean flow transformation, offering improved accuracy over conventional LoW-based models.

This study employs interface-resolved direct numerical simulations to investigate the settling characteristics of non-spherical particles in homogeneous isotropic turbulence, focusing on the effect of particle shape on settling dynamics. Three particle shapes are considered and compared: spherical particles, ellipsoidal particles with aspect ratio X = 2, and cylindrical particles with aspect ratio X = 1 and X = 2. In addition, the effect of the Galileo number (Ga) is considered. We released multiple particles at a low volume fraction (0.1%) to gather robust statistics while minimizing interactions, in order to mimic the settling of an isolated particle. Results show that the particle shape plays a critical role in the orientation dynamics: ellipsoidal particles exhibit a pronounced preferential orientation with the major axis perpendicular to the direction of gravity that intensifies with increasing Ga, while cylindrical particles (X = 1) remain randomly oriented. Notably, a bifurcation is observed for cylindrical particles (X = 2) as Ga increases: they maintain random orientations at low Ga but transition to a preferential orientation with the major axis perpendicular to the direction of gravity at high Ga, revealing a nonlinear coupling between gravitational forcing and particle elongation. The mean settling velocity reveals a clear ordering: spherical particles settle the fastest, followed by ellipsoidal particles, while cylindrical particles settle the slowest. We find that this reduction in settling velocity for non-spherical particles arises from an increased pressure drag and the greater viscous dissipation associated with the unsteady rotational motion. Furthermore, the ordering of settling velocities in the turbulent flow is found to differ from that observed in a quiescent fluid. These findings further elucidate the influence of particle shape on turbulence-particle interactions and subsequently the settling velocity of non-spherical particles in turbulent flows.

An advanced super-resolution reconstruction method for turbulent flows should be efficient, avoiding reliance on high-resolution data; reliable, strictly adhering to physical laws; and accurate, effectively reconstructing both small- and large-scale flow structures. To achieve these objectives, we developed TurbRec, an unsupervised, physically-consistent full-scale turbulent reconstruction model. This novel framework is designed to capture both high- and low-frequency information for precise reconstruction of small- and large-scale turbulent structures. TurbRec relies entirely on low-resolution inputs and governing equations, eliminating the need for high-resolution data and ensuring both efficiency and physical fidelity. Inspired by the multi-scale nature of turbulence and the energy cascade process, our approach integrates frequency-aware Fourier feature embeddings, capturing the full spectrum of turbulent frequencies and enhancing the model’s ability to represent intricate spectral details. To maintain physical consistency, we embed the governing equations within the super-resolution process and introduce a series of physics-motivated strategies, including segmented learning with overlapping windows, multi-scale normalization, and an enhanced residual learning network architecture. These innovations address known challenges in turbulent flow reconstruction, such as accurately predicting small-scale structures and ensuring smooth transitions across temporal boundaries. Numerical experiments demonstrate that TurbRec consistently outperforms traditional physics-informed neural networks in predicting instantaneous spatial flow structures, turbulent kinetic energy spectra, and various turbulent statistics.

Semivolatile and intermediate-volatility organic compounds (S/IVOCs) are key but under-constrained precursors of urban secondary organic aerosol (SOA). We leveraged the COVID-19 lockdown-to-reopening transition in Beijing as a natural experiment to track molecularly resolved gaseous organics using comprehensive two-dimensional gas chromatography-mass spectrometry (GC × GC-MS). Across 305 quantified species, IVOCs accounted for ∼37% of the measured gas-phase loading yet dominated diagnosed incremental SOA formation (∼89%) in a simplified observation-constrained 0D diagnostic implementation based on two-dimensional volatility basis set (2D-VBS) concepts. Partial least squares discriminant analysis (PLS-DA) revealed compositional shifts, with n-C8, 2-butyl-1-octanol, and hexamethylcyclotrisiloxane (D3) enriched after reopening, indicating rebounds in traffic and volatile chemical product-related emissions; within the VMS mixture, D3 became dominant while lockdown-phase D3 remained near the low-end quantification regime. Meteorological normalization indicated that the decline in total gas-phase organics could not be explained by local meteorology alone and was consistent with a weakened regional background. Despite stronger anthropogenic tracers after reopening, diagnosed daily SOA yields decreased. This efficiency drop appears more strongly associated with thermodynamic constraints than with a broad reduction in atmospheric oxidation capacity. These results indicate that IVOC-enriched noncombustion and evaporative emissions may materially influence diagnosed urban SOA formation within the measured gas-phase precursor pool.

Determining the particle chemical morphology is crucial for unraveling reactive uptake in atmospheric multiphase and heterogeneous chemistry. However, it remains challenging due to the complexity and inhomogeneity of aerosol particles. Using a scanning transmission X-ray microscopy (STXM) coupled with near-edge X-ray absorption fine structure (NEXAFS) spectroscopy and an environmental cell, we imaged and quantified the chemical morphology and hygroscopic behavior of individual submicron urban aerosol particles. Results show that internally mixed particles composed of organic carbon and inorganic matter (OCIn) dominated the particle population (73.1±7.4 %). At 86 % relative humidity, 41.6 % of the particles took up water, with OCIn particles constituting 76.8 % of these hygroscopic particles. Most particles exhibited a core-shell structure under both dry and humid conditions, with an inorganic core and an organic shell. Our findings provide direct observational evidence of the core-shell structure and water uptake behavior of typical urban aerosols, which underscore the importance of incorporating the core-shell structure into models for predicting the reactive uptake coefficient of heterogeneous reactions.

With the increasing demand for rapid aerodynamic evaluations in the automotive industry, data-driven surrogate models for predicting vehicle surface pressure have emerged as promising alternatives to traditional computational fluid dynamics (CFD) simulations. However, most existing models fall short in capturing the intrinsic spatial correlations of complex 3D vehicle geometries, particularly when represented as unstructured point clouds. To address these limitations, we propose the spatially-aware transformer operator (SATO), a novel neural operator framework that unifies global and local spatial correlation modeling through two complementary modules: physics-attention and serialized-attention. SATO introduces a spatial aggregation mechanism to capture large-scale geometric structures while simultaneously employing a serialization technique based on space-filling curves to transform unstructured 3D points into structured sequences, allowing efficient local feature extraction. This dual-attention mechanism enables SATO to achieve multi-scale feature fusion with nearly linear computational complexity. Extensive evaluations are conducted on two datasets: the ShapeNet Car dataset, containing a wide variety of simplified vehicle shapes, and the DrivAerNet dataset, comprising industrial-grade, high-fidelity car models. Results demonstrate that SATO reduces the relative L2 error in pressure prediction by 13 % and 11 %, respectively, compared to state-of-the-art methods, while maintaining real-time inference speed (under one second) for vehicles with over 0.42 million mesh vertices. The complementarity of the two attention mechanisms is further substantiated through a visual analysis of their internal operations. Our study highlights the effectiveness of integrating global and local spatial correlations in transformer-based operator learning and establishes SATO as a robust surrogate model for real-time aerodynamic evaluations of arbitrary vehicle geometries.

In scientific computing, the burgeoning usage of physics-informed neural networks (PINNs) for solving partial differential equations(PDEs) has spurred the need for, and ongoing research into, more accurate and efficient PINNs. One bottleneck of current PINNs is the computation of the high-order derivatives via automatic differentiation which often necessitates substantial computing resources especially when dealing with complex PDEs and high dimensional problems. To tackle this, we propose a spectral-based neural network that substitutes the differential operator with a multiplication. Compared to PINNs, our framework requires less GPU memory and a shorter training time. Furthermore, the exponential convergence of the spectral basis makes our approach more accurate. Moreover, to handle the different situations between the physics domain and the spectral domain, we provide a strategy to efficiently train networks in spectral domain. Through a series of comprehensive experiments, we validate the aforementioned merits of our proposed network.

Counter-rotating propellers have gained increasing attention in unmanned aerial vehicle applications due to their load capacity and redundancy. However, interactions between propellers lead to variations in the noise emission. Edgewise flight generates more complex flow fields and blade-vortex interactions, which significantly alter propeller aerodynamics and aeroacoustics. This work numerically investigates the aerodynamic performance and aeroacoustic characteristics of counter-rotating propellers during edgewise flight. A high-fidelity delayed detached-eddy simulation within the OpenFOAM framework is utilized to investigate a two-blade counter-rotating propeller system undergoing edgewise flight with different edgewise speeds. The contribution of upstream and downstream propeller blades to noise generation and the directivity of the noise radiation are investigated separately. Detailed analyses of the aerodynamic interactions between the propellers and the aeroacoustic source mechanisms are also evaluated. The findings provide critical insights into the aerodynamics and noise characteristics of counter-rotating propellers undergoing edgewise flight, which are of great importance for both performance optimization and noise reduction.

Partial differential equations (PDEs) encode fundamental physical laws, yet closed-form analytical solutions for many important equations remain unknown and typically require substantial human insight to derive. Existing numerical, physics-informed, and data-driven approaches approximate solutions from data rather than systematically deriving symbolic expressions directly from governing equations. Here we introduce LawMind, a law-driven symbolic discovery framework that autonomously constructs closed-form solutions from PDEs and their associated conditions without relying on data or supervision. By integrating structured symbolic exploration with physics-constrained evaluation, LawMind progressively assembles valid solution components guided solely by governing laws. Evaluated on 100 benchmark PDEs drawn from two authoritative handbooks, LawMind successfully recovers closed-form analytical solutions for all cases. Beyond known solutions, LawMind further discovers previously unreported closed-form solutions to both linear and nonlinear PDEs. These findings establish a computational paradigm in which governing equations alone drive autonomous symbolic discovery, enabling the systematic derivation of analytical PDE solutions.

Historically, mapping the skin-friction coefficient $C_f$ and momentum-thickness Reynolds number $Re_\theta$ onto their 'incompressible' counterparts governed by established scaling laws has been widely used for efficient prediction of surface drag in zero-pressure-gradient compressible turbulent boundary layers. However, reassessment using comprehensive direct numerical simulation databases reveals that existing formulations exhibit physical inconsistencies and fail to provide consistently accurate predictions. To address these limitations, we establish a general framework for skin-friction transformations derived as a direct consequence of mean velocity mappings. An a priori analysis shows that while the framework itself is mathematically rigorous, the limited accuracy of existing mean velocity scalings in the outer region imposes intrinsic constraints that remain unavoidable. To further facilitate predictive applications, we reformulate and unify van Driest’s classical theory via semi-analytical derivations. We show that the historical success of the van Driest II transformation does not stem from a superior physical description, but from a fortuitous cancellation of errors in traditional asymptotic approximations, which inherently fail at practical Reynolds numbers. Abandoning these approximations, modified skin-friction transformations based on exact integral formulations are evaluated through both a priori scaling-law collapse and a posteriori prediction of $C_f$, exhibiting performance consistent with their underlying compressible laws of the wall. In particular, a novel scaling incorporating an established velocity transformation (Volpiani et al., 2020, Phys. Rev. Fluids, vol. 5, 052602) provides the best practical compromise currently available, achieving the highest predictive accuracy across a broad range of free-stream Mach numbers ($0.30 \leq M_\infty \leq 13.64$) and wall-to-recovery-temperature ratios ($0.18 \leq \bar T_w / T_r \leq 1.89$), with maximum and mean relative errors in $C_f$ of 8.38% and 2.61%, respectively. Moreover, this skin-friction scaling can be seamlessly embedded within a recently proposed mean-flow prediction framework (Ying et al., 2025, J. Fluid Mech., vol. 1021, A30), enabling the simultaneous reconstruction of complete mean velocity and temperature profiles. Ultimately, by clarifying their underlying theoretical homology with contemporary inverse-velocity-transformation frameworks, this work elevates skin-friction transformations from heuristic correlations into a rigorously formulated physical paradigm.

α-Hydroxy hydroperoxides (α-HHPs) are an important class of atmospheric peroxides, yet their atmospheric fate remains poorly characterized due to limited observations and a lack of modeling studies. Here, we selected hydroxymethyl hydroperoxide (HMHP), the simplest α-HHP, as a proxy for detailed investigation. Field observations were conducted in Beijing during summer and winter 2024 to characterize HMHP's atmospheric behavior and its correlations with other species. HMHP exhibits distinct diurnal variations compared to other common peroxides, likely due to its unique formation pathway via alkene ozonolysis. We also found its correlation with PM2.5 and O3 differs between seasons. In summer, the box model simulation significantly overestimated HMHP concentrations when only gas phase loss processes were considered. Adding heterogeneous uptake driven by S(IV) oxidation and Fenton-like reactions on wet aerosols substantially improved model performance. These aerosol-phase processes further contribute to particle growth and aging by forming inorganic and organic components. In winter, applying the summer-improved mechanism led to significant underestimation on HMHP during PM2.5 growth episodes. We propose that the heterogeneous reaction between H2O2 and HCHO—while playing a minor role in summer when aerosols are wet—becomes important in winter when aerosols are dry and reactive surface sites are available. Overall, aerosols play a dual role: they act as a sink for HMHP through uptake on wet aerosols, but as a source through surface reactions on dry aerosols. Based on our simulation, in summer, HMHP is sourced mainly from Criegee intermediates chemistry and lost via heterogeneous uptake (over 50%), whereas in winter, heterogeneous reactions on dry aerosols become an important source, with dry deposition and uptake as the main sinks. Our findings fill a critical gap in previous research on the observation and modeling study of α-HHPs and reveal a close link between these peroxides and particulate matter.

Reliable extrapolation remains a central challenge for generative models in computational physics, because models trained over finite ranges of time, parameters, or geometries may produce physically inconsistent predictions outside the training distribution. We introduce a least-action-principle-guided diffusion, LAPG, a framework that promotes physical consistency during inference rather than relying solely on constraints imposed during training. The method combines a conditional score-based diffusion model with an action-derived physical guidance score. In the first stage, the learned score model generates an in-distribution proposal; in the second, an action-based variational prior refines this proposal toward the target out-of-distribution condition. This formulation turns the principle of least action into a differentiable inference-time correction mechanism and provides an alternative to pointwise residual penalties that often require empirical loss balancing. We evaluate LAPG on representative ordinary- and partial-differential-equation systems, including free fall, conservative and dissipative spring-mass dynamics, interacting point vortices, and potential flow over parameterized airfoils. In temporal, parameter, and geometric extrapolation tests, LAPG reduces phase drift, preserves dissipative decay, captures vortex motion, and improves the lift response of airfoil flows compared with training-time physics-informed baselines.

External body forces influence the evolution of the distribution function in mesoscopic methods through the particle-velocity-space gradient term in the Boltzmann equation. Simplified forcing models for continuum flows can be categorized into moment-expansion-based and Chapman-Enskog-expansion-based models, while their performance in compressible turbulence remains insufficiently explored. Using the partial internal energy double-distribution-function (DDF) framework, this study addresses that gap. We systematically derive the moment constraints for any forcing model to recover the Navier-Stokes-Fourier equations with arbitrary bulk viscosity for the first time. Within the partial internal energy DDF framework, we novelly propose a moment-expansion-based forcing model for compressible flows, which accounts for the higher-order structure of the distribution function without requiring increased Gauss-Hermite quadrature accuracy. For comparison, we also consider two classical Chapman-Enskog-based models, the He model [He et al., J. Comput. Phys. 146, 282 (1998)], which requires higher-order quadrature, and the Kupershtokh model [Kupershtokh et al., Comput. Math. Appl. 58, 965 (2009)], which neglects higher-order structural effects, even though both models satisfy the same velocity moment constraints. All three models yield stable simulations of forced compressible turbulence using partial internal energy DDF-based discrete unified gas kinetic scheme. However, nonlinear interactions lead to observable, though small, differences in turbulence statistics under identical initial conditions. Overall, the results indicate that each model is capable of providing reasonable turbulence statistics.

The ingestion of turbulent inflow by propellers generates broadband noise, which constitutes one of the primary aeroacoustic sources in propeller systems. This study employs a synthetic turbulence method to generate homogeneous isotropic turbulence with controllable parameters, featuring varying length scales and turbulence intensities. Numerical simulations are conducted to investigate the interaction between the turbulence and the propeller. Impact of turbulence length scales and intensities are evaluated separately. Turbulence ingestion noise emerges as the dominant noise source across all operational conditions. When ingesting turbulent inflow, the aeroacoustic directivity pattern become rotational axis dominant, while the noise spectra features a "haystack" pattern. Then, the aeroacoustic sources and blade-to-blade correlation are further examined. Key findings indicate that while larger turbulence scales induce stronger blade correlation, they generate reduced high-frequency noise components, consequently yielding lower far-field acoustic radiation.

We perform numerical simulations of forced homogeneous isotropic turbulence over a range of bulk viscosities, Reynolds numbers and Mach numbers to investigate the scaling of key flow statistics. Using the Helmholtz decomposition, we analyse the scalings of Favre-averaged turbulent kinetic energy (TKE), root-mean-square (r.m.s.) pressure, pressure dilatation, dilatational dissipation and higher-order velocity-gradient moments. Additionally, new models are proposed for the pressure-dilatation term and the bulk-viscosity dependence of dilatational dissipation. Although the solenoidal and dilatational components of the Favre-averaged TKE are not strictly orthogonal, our numerical results demonstrate that their ratio is well approximated by the squared ratio of the corresponding r.m.s. velocities. The r.m.s. pressure approaches the pseudo-sound scaling as bulk viscosity increases. Within the Donzis r.m.s. pressure model (Donzis & John 2020 Phys. Rev. Fluids 5(8), 084609), we find that the solenoidal contribution becomes dominant for large bulk viscosity. Pressure dilatation is found to depart systematically from pseudo-sound predictions: without bulk viscosity it favours transfer from kinetic to internal energy, while finite bulk viscosity can reverse this transfer at high Mach numbers. The scaling exponent of dilatational dissipation is shown to vary with bulk viscosity, enabling a corrected model for its exponent and prefactor. Velocity-gradient skewness and flatness reveal that the onset of shocklet-induced divergence is delayed with increasing bulk viscosity and may be suppressed entirely. The results extend recent velocity-ratio-based scaling frameworks and provide modelling insights into compressible turbulence.

The hydroxyl radical (OH) plays a central role in atmospheric chemistry, however, its accurate measure-ment by laser induced fluorescence with gas expansion (LIF-FAGE) is unavoidably compromised by wavelength drift of the excitation laser. To overcome this limitation, a real time reference system for active wavelength locking has been developed and systematically characterized in this work. Stable, and high concentration OH radicals were generated through thermolysis of ambient air on a heated filament within a low pressure cell. The excitation source was a308 nm laser produced by frequency doubling the output of a DCM-ethanol dye laser pumped by an Nd:YAG laser. The induced fluorescence was monitored in real time us-ing a non-gated photomultiplier tube (PMT). The wavelength locking program, implemented with a closed loop feedback mechanism, dynamically adjusted the laser wavelength to the optimal OH excitation line. Through comprehensive characterization of key parameters, including laser power, filament operating conditions (current, voltage), and cell environment (pressure/inlet flow rate, inlet gas relative humid-ty), an optimal operational window of the reference system has been identified. A 12 h continuous measurement demonstrated high system stability in OH generation and detection, the observed fluorescence intensity showed a low drift rate of0.2 % h-1during the first nine hours. The good robustness of the reference system, and its integrated wavelength locking program, enable long-term and accurate ambient OH radical quantification in LIF-FAGE measurements

A long-standing route to efficient surface-drag prediction in zero-pressure-gradient compressible turbulent boundary layers is to map the skin-friction coefficient C_f and momentum-thickness Reynolds number Re_θ onto their `incompressible' counterparts. Reassessment against an extensive DNS database shows that existing formulations do not consistently recover the reference incompressible skin-friction behaviour, even when transformed data exhibit improved collapse. We define the mapped `incompressible' state as a constant-property counterpart of the physical compressible boundary layer and derive the transformation factors from prescribed mean-velocity and wall-normal-coordinate mappings. This definition-first approach links skin-friction scaling to the full-layer accuracy of the underlying velocity transformation and exposes inherited outer-layer errors. Van Driest's theory is recast in a finite-Re exact-integral form, with the classical vD I and II transformations recovered as leading-order asymptotic reductions. Their limitations at finite Reynolds numbers are quantified, and the historical success of vD II is traced to a fortuitous cancellation of truncation errors. The exact-integral formulation then yields modified transformations assessed through a priori scaling and standalone a posteriori prediction of C_f from prescribed macroscopic and wall-thermal inputs. The VIPL-based modified transformation gives the best overall performance. Across 0.30 ≤ M_∞≤ 13.64 and -0.55 ≤≤ 2.85, its prediction errors remain below 11%, with a mean error of 3.07%. Overall, the analysis places skin-friction transformations on a mapping-based exact-integral footing, relating them directly to prescribed mean-flow mappings while avoiding the leading-order asymptotic truncations that limit classical van Driest theory at finite Reynolds numbers.