Microscopic Velocity Research Articles

Physics-informed neural networks for weakly compressible flows using Galerkin–Boltzmann formulation

In this work, we study the Galerkin–Boltzmann formulation within a physics-informed neural network (PINN) framework to solve flow problems in weakly compressible regimes. The Galerkin–Boltzmann equations are discretized with second-order Hermite polynomials in microscopic velocity space, which leads to a first-order conservation law with six equations. Reducing the output dimension makes this equation system particularly well suited for PINNs compared with the widely used D2Q9 lattice Boltzmann velocity space discretizations. We created two distinct neural networks to overcome the scale disparity between the equilibrium and non-equilibrium states in collision terms of the equations. We test the accuracy and performance of the formulation with benchmark problems and solutions for forward and inverse problems with limited data. We compared our approach with the incompressible Navier–Stokes equation and the D2Q9 formulation. We show that the Galerkin–Boltzmann formulation results in similar L2 errors in velocity predictions in a comparable training time with the Navier–Stokes equation and lower training time than the D2Q9 formulation. We also solve forward and inverse problems for a flow over a square, try to capture an accurate boundary layer, and infer the relaxation time parameter using available data from a high-fidelity solver. Our findings show the potential of utilizing the Galerkin–Boltzmann formulation in PINN for weakly compressible flow problems.

Calculation Model for the Exit Decision Sight Distance of Right-Turn Ramps on the Left at Interchange

Given the rapid construction of freeways in developing countries such as China, land use is constantly under strict constraints, leading to challenges in adopting conventional layouts for interchanges. Implementing right-turn ramps on the left (RTRL) at interchanges can minimize land occupancy; however, the traffic safety level in this type of diversion area design requires extra attention. This study examines the decision sight distance for right-turn exit ramps on the left side. Utilizing unmanned aerial vehicle (UAV) video and the YOLOv3 target detection algorithm, the original trajectory data of vehicles in the diversion area is extracted. Employing Kalman filtering and Frenet coordinate system conversion reveals microscopic vehicle lane-change patterns, velocities, and time headways. Furthermore, the driving simulation experiment assesses driver behaviors in RTRL, with subjective, task performance, and physiological measure indicators. Ultimately, the range of the decision sight distance is defined, and establishing a calculation model involves determining relevant parameters based on measured data and simulation outcomes. The results indicate potential insufficiencies in the decision sight distance when standardized values are applied to RTRL.

A hierarchy of the optimal velocity model with optimal path for pedestrian evacuation: From microscopic to macroscopic models

We study the evacuation process of pedestrians by adopting an optimal velocity model. The Eikonal equation is coupled to the optimal velocity model for optimal path to guide the movement directions of pedestrians. It depends on pedestrian density. We derive the corresponding mean field equation, hydrodynamic and scalar models from the scaled microscopic optimal velocity model. Several numerical experiments are performed in a corridor with two exits. We show and compare results on the microscopic as well as on the hydrodynamic and scalar models. Results from microscopic model are closed to the hydrodynamic and scalar models when a large number of particles are considered in microscopic simulation. The computation time increases as number of particles in microscopic simulation increases. The computation times of the hydrodynamic and scalar models are less than the computation time of the microscopic model with large number of particles. Hence it is beneficial to apply the hydrodynamic and scalar models over the microscopic model when a large number of particles in microscopic system are considered.

Local well‐posedness of classical solutions for the kinetic self‐organized model of Cucker–Smale type

We study in this paper the local well‐posedness of classical solutions to the Cauchy problem for a kinetic self‐organized model of Cucker–Smale type for collective motions, which includes two cases of velocity alignment mechanisms: normalized and nonnormalized reorientation cases. The main concern is the a priori estimates for both cases. Our treatments rely on two key ingredients: One point is to deal with the possible singularity and high nonlinearity arising from the normalized/nonnormalized mean velocity, and the other is to control the growth with respect to the microscopic velocity variable in the whole space by employing weighted estimates.

Numerical Equivalent Acoustic Material for Air-Filled Porous Absorption Simulations in Finite Different Time Domain Methods: Design and Comparison

Extracting the microscopic parameters of a porous material is a complex task, and attempts have been made to develop models that can simulate their characteristics, gathering the least amount of information possible. As a case in point, tests to evaluate macroscopic behaviors such as tortuosity, which depends directly on the microscopic fluid velocities, are highly susceptible to generate errors if the precision of the measurement devices is not correct, and the same goes for the other parameters. For this reason, in this paper, a sound propagation model in porous materials with a rigid frame is presented based on a local theory, which tries to simplify, even more, the way to obtain the basic characteristics of porous materials, such as their absorption coefficient at normal and random incidence, and their normal surface impedance. The proposed linearized equivalent fluid model presents four phenomenological coefficients, which characterize acoustic propagation trough the material. Their values are obtained from the material thickness and a measurement in an impedance tube following the ISO 10534 standard. Thus, what is only required is the measured absorption coefficient, either on one third or one octave bands, to fully represent the acoustic behavior in the finite different in time domain (FDTD) method. The model has been simulated with FDTD in porous and fibrous kernels, and the results show a strong agreement with the laboratory measurements and with the analytical results calculated with well-established semi-phenomenological models.

Direct numerical simulation of supersonic nanoparticles flow in free-molecule regime using the angular coefficient method

We employ molecular flow methods to numerically simulate the supersonic nanoparticles flow in free-molecule regime. To streamline the computational complexity, interaction forces between the gas and solid particles are disregarded. We first develop a discrete phase model (DPM) method that integrates the non-rigid body collision model, enabling an accurate simulation of nanoparticle diffusion under the influence of the drag force and Brownian motion force. The nanoparticles considered in this study have sizes below 10 nm, and the accuracy of the DPM method is verified by comparing its results with experimental data. Subsequently, we theoretically and numerically investigate the transmission probability and number density of N2 molecules flowing through two-dimensional (2D) channels and three-dimensional (3D) tubes by using the angular coefficient (AC) method and the direct simulation Monte Carlo (DSMC) method. The findings indicate that as the diameter of the nanoparticle (dp) decreases to 1 nm, the diffusion coefficient (D) and the root mean square displacement (x) of nanoparticles approach the N2 molecules. The microscopic velocity of most N2 molecules falls within the range of 62–1400 m/s, and the macroscopic velocity of N2 flow falls within the range of Ma = 1.28–1.35. In contrast to the DSMC method, the AC method exhibits enhanced accuracy even with a reduced number of grids and obviates the process for large-scale sampling. Additionally, the solution time required by the AC method is approximately 1/10 and 1/13–1/32 of the DSMC method in 3D cylindrical tubes and 2D channels, respectively. Moreover, the AC method demonstrates superior adaptability when dealing with complex geometries.

On the yield criterion of porous materials by the homogenization approach and Steigmann–Ogden surface model

In this work, we investigate the yield criterion of nanoporous materials by using homogenization approach and Steigmann–Ogden surface model. The representative volume element is proposed as an infinite matrix containing a tiny nanovoid. The matrix is incompressible, rigid-perfectly plastic, von Mises materials and nanovoids are dilute and equal in size. First, the constitutive of microscopic stress and microscopic strain rate is established based on the flow criterion. Secondly, according to the Hill’s lemma, the relationship between the macroscopic equivalent modulus and the microscopic equivalent modulus is established by homogenization approach. Thirdly, the macroscopic equivalent modulus containing the Steigmann–Ogden surface model including surface parameters, porosity and nanovoid radius is derived from the trial microscopic velocity field. Finally, an implicit macroscopic yield criterion for nanoporous materials is developed. For surface modulus, nanovoids radius and porosity studies are developed through extensive numerical experiments. The research results in this paper have reference significance for the design and manufacture of nanoporous materials.

Polytropic representation of non-isotropic kinetic pressure tensor for non-ideal plasma fluids in relativistic jets

Non-ideal fluids are likely to be affected by the occurrence of pressure anisotropy effects, whose understanding for relativistic systems requires knowledge of the energy–momentum tensor. In this paper, the case of magnetized jet plasmas at equilibrium is considered, in which both microscopic velocities of constituent particles and the continuum fluid flow are treated as relativistic ones. A theoretical framework based on covariant statistical kinetic approach is implemented, which permits the proper treatment of single-particle and phase-space kinetic constraints and, ultimately, the calculation of the system continuum fluid fields associated with physical observables. A Gaussian-like solution for the kinetic distribution function (KDF) is constructed, in which the physical mechanism responsible for the generation of temperature anisotropy is identified with magnetic moment conservation. A Chapman–Enskog representation of the same KDF is then obtained in terms of expansion around an equilibrium isotropic Juttner distribution. This permits the analytical calculation of the fluid 4-flow and stress–energy tensor and the consequent proof that the corresponding kinetic pressure tensor is non-isotropic. As a notable result, the validity of a polytropic representation for the perturbative non-isotropic pressure contributions is established, whereby directional pressures exhibit specific power-law functional dependences on fluid density.

Low-Gradient Magnetophoresis of Nanospheres and Nanorods through a Single Layer of Paper.

The possible magnetophoretic migration of iron oxide nanoparticles through the cellulosic matrix within a single layer of paper is challenging with its underlying mechanism remained unclear. Even with the recent advancements of theoretical understanding on magnetophoresis, mainly driven by cooperative and hydrodynamics phenomena, the contributions of these two mechanisms on possible penetration of magnetic nanoparticles through cellulosic matrix of paper have yet been proven. Here, by using iron oxide nanoparticles (IONPs), both nanospheres and nanorods, we have investigated the migration kinetics of these nanoparticles through grade 4 Whatman filter paper with a particle retention of 20-25 μm. By performing droplet tracking experiments, the real-time stained area growth of the particle droplet on the filter paper, under the influences of a grade N40 NdFeB magnet, were recorded. Our results show that the spatial and temporal expansion of the IONP stain is biased toward the magnet and such an effect is dependent on (i) particle concentration and (ii) particle shape. The kinetics data were first analyzed by treating it as a radial wicking fluid, and later the IONP distribution within the cellulosic matrix was investigated by optical microscopy. The macroscopic flow front velocities of the stained area ranged from 259 μm/s to 16 040 μm/s. Moreover, the microscopic magnetophoretic velocity of nanorod cluster was also successfully measured as ∼214 μm/s. Findings in this work have indirectly revealed the strong influence of cooperative magnetophoresis and the engineering feasibility of paper-based magnetophoretic technology by taking advantage of magnetoshape anisotropy effect of the particles.

Quantifying fabric anisotropy of granular materials using wave velocity anisotropy: a numerical investigation

Fabric anisotropy is a sought-after micro index to correlate macro mechanical responses of granular materials. In this work, the discrete-element method is utilised to simulate multi-directional bender element tests in granular soils to obtain the evolution of wave velocities during drained conventional triaxial (CT) and true triaxial (TT) tests; the contact normal-based fabric is simultaneously monitored for bridging the fabric anisotropy and wave velocity anisotropy. The results show that stress-normalised wave velocities and microscopic fabric, including contact normal distribution and coordination number, remain nearly constant until a stress ratio threshold is reached. After the threshold value is reached, stress-normalised wave velocities start to decrease, especially in the minor principal stress direction, accompanied by significant adjustment of coordination number and fabric anisotropy. The results also reveal that the normalised wave velocity depends on the contact normal densities in the wave propagation and particle oscillation directions. With the contact normal distribution represented by a density function, a good linear relationship between the microscopic fabric anisotropy and macroscopic wave velocity anisotropy is obtained for both CT and TT tests.

Numerical schemes for a multi-species BGK model with velocity-dependent collision frequency

We consider a kinetic description of multi-species gas mixture modeled with Bhatnagar-Gross-Krook (BGK) collision operators, in which the collision frequency varies not only in time and space but also with the microscopic velocity. In this model, the Maxwellians typically used in standard BGK operators are replaced by a generalization of such target functions, which are defined by a variational procedure [20]. In this paper we present a numerical method for simulating this model, which uses an Implicit-Explicit (IMEX) scheme to minimize a certain potential function, mimicking the Lagrange functional that appears in the theoretical derivation. We show that theoretical properties such as conservation of mass, total momentum and total energy as well as positivity of the distribution functions are preserved by the numerical method, and illustrate its usefulness and effectiveness with numerical examples.

Application of phase field model coupled with convective effects in binary alloy directional solidification and roll casting processes

Based on the Kim-Kim-Suzuki (KKS) phase field model coupled with the thermodynamic parameters, the transformation process from columnar dendrites to equiaxed crystals during directional solidification of aluminium alloy was simulated, and the effects of phase field parameters on the growth morphology and dendrite segregation were discussed. Furthermore, considering the effect of the microcosmic flow field, the convection influence gradient term is introduced into KKS formula near the solid-liquid interface, and the phase field model considering flow field was applied to the inherent convective environment of the actual roll casting process, also the multiple dendrites growth behavior of magnesium alloy under the action of microscopic convection was further explored. When coupling calculation of microscopic velocity field and pressure field, the staggered grid method was used to deal with the complex interface. The combined solution of Marker in Cell (MAC) algorithm and phase field discrete calculation was realized. In order to further describe the influence of convection on the solidification process, the roll casting experiments are used to verify the impact growth of multiple dendrites under convection. The results show that the dendrites undergo solute remelting and the dendrites melt into equiaxed crystals, showing the phenomenon of Columnar to Equiaxed Transition (CET).

Integrated Velocity Prediction Method and Application in Vehicle-Environment Cooperative Control Based on Internet of Vehicles

Rapid progress has been gained in the field of advanced communication technologies, which also promote parallel developments in the Internet of Vehicles (IoVs). In this context, vehicle-environment cooperative control can be integrated into next-generation vehicles to further improve the vehicle's performance, in particular energy efficiency. Accurate prediction of future velocity profiles on basis of IoVs can be a critical breakthrough, which can contribute much to vehicle operation efficiency promotion. In this paper, an integrated velocity prediction (IVP) method fully taking advantage of IoVs is proposed and demonstrated through a case study. In the IVP method, both the macroscopically and microscopically predicted velocity profiles are considered. The macroscopic velocity profiles are predicted via traffic flow analysis (TFA) in multi-access edge computing units (MECUs) which are situated alongside the route. Microscopic velocity profiles are forecasted through Mondrian forest (MF) algorithm in the on-board vehicle control unit (VCU). Final velocity prediction is generated through combination of the macroscopic and microscopic profiles in frequency domain in on-board VCU through fast Fourier transform (FFT) and inverse FFT. A case study validates the distinguished performance of IVP method and demonstrates its significant contribution to vehicle performance improvement.

Effect of individual differences on the jamming transition in traffic flow.

The individual difference, particularly in drivers' distance perception, is introduced in the microscopic one-dimensional optimal velocity model to investigate its effect on the onset of the jamming instability seen in traffic systems. We show analytically and numerically that the individual difference helps to inhibit the traffic jam at high vehicle densities while it promotes jamming transition at low vehicle densities. In addition, the jamming mechanism is further investigated by tracking how the spatial disturbance travels through traffics. We find that the jamming instability is uniquely determined by the overall distribution of drivers' distance perception rather than the spatial ordering of vehicles. Finally, a generalized form of the optimal velocity function is considered to show the universality of the effect of the individual difference.

Dissipation Mechanisms for Fluids and Objects in Relative Motion Described by the Navier-Stokes Equation.

This work demonstrates that an additional resistance term should be included in the Navier–Stokes equation when fluids and objects are in relative motion. This is based on an observation that the effect of the microscopic molecular random velocity component parallel to the macroscopic flow direction is neglected. The two components of the random velocity perpendicular to the local mean flow direction are accounted for by the viscous resistance, e.g., by Stokes’ law for spherical objects. The relationship between the mean- and the random velocity in the longitudinal direction induces differences in molecular collision velocities and collision frequency rates on the up- and downstream surface areas of the object. This asymmetry therefore induces flow resistance and energy dissipation. The flow resistance resulting from the longitudinal momentum transfer mode is referred to as thermal resistance and is quantified by calculating the net difference in pressure up- and downstream the surface areas of a sphere using a particle velocity distribution that obeys Boltzmann’s transport equation. It depends on the relative velocity between the fluid and the object, the number density and the molecular fluctuation statistics of the fluid, and the area of the object and the square root of the absolute temperature. Results show that thermal resistance is dominant compared to viscous resistance considering water and air in slow relative motion to spherical objects larger than nanometer-size at ambient temperature and pressure conditions. Including the thermal resistance term in the conventional expression for the terminal velocity of spherical objects falling through liquids, the Stokes–Einstein relationship and Darcy’s law, corroborates its presence, as modified versions of these equations fit observed data much more closely than the conventional expressions. The thermal resistance term can alternatively resolve d’Alembert’s paradox as a finite flow resistance is predicted at both low and high relative fluid–object velocities in the limit of vanishing fluid viscosity.

Experimental investigation on the macroscopic spray and microscopic droplet diameter, velocity and temperature of R404A flashing spray

Two-phase flashing spray elicits wide interest in scientific and industrial fields. It can produce fine atomization with small size and low temperature droplets because of explosive atomization and strong evaporation. This study conducted an experimental investigation on the macroscopic spray and microscopic droplet diameter, velocity, and temperature through various measurement methods with a low saturation-temperature refrigerant R404A, which holds a special interest in laser dermatology. The spray presented a bowl spray configuration with rapid expansion at the nozzle exit. A core spray area near the nozzle field was revealed, in which the droplet Sauter mean diameter (D32) and temperature (Td) decreased rapidly while the droplet average velocity (V) exhibited an obvious acceleration with increasing axial distance. All the number distributions of droplet diameter and velocity presented Gaussian shapes. The cumulative distribution of droplet diameter followed the Rosin–Rammler correlation perfectly. The uniformity constant and characteristic droplet size were proposed for various axial and radial positions. The radial distributions of D32, V, and Td at various spray axial sections presented different profiles, among which the non-dimensional profiles of V showed a self-similar pattern. The two-dimensional thermal field presented a contractive pattern and a narrow spray width after the rapid expansion at the nozzle exit, which helped to control the spray cooling area more precisely compared with other expansion-like sprays with larger spray width. Experimental data could also provide validation data for the numerical simulation of flashing spray.

High accuracy capillary network representation in digital rock reveals permeability scaling functions

Permeability is the key parameter for quantifying fluid flow in porous rocks. Knowledge of the spatial distribution of the connected pore space allows, in principle, to predict the permeability of a rock sample. However, limitations in feature resolution and approximations at microscopic scales have so far precluded systematic upscaling of permeability predictions. Here, we report fluid flow simulations in pore-scale network representations designed to overcome such limitations. We present a novel capillary network representation with an enhanced level of spatial detail at microscale. We find that the network-based flow simulations predict experimental permeabilities measured at lab scale in the same rock sample without the need for calibration or correction. By applying the method to a broader class of representative geological samples, with permeability values covering two orders of magnitude, we obtain scaling relationships that reveal how mesoscale permeability emerges from microscopic capillary diameter and fluid velocity distributions.

Goodness-of-fit test for isochrone fitting in the Gaia era

Context. The increasing precision in observational data made available by recent surveys means that the reliability of stellar models can be tested. For this purpose, a firm theoretical basis is crucial for evaluating the agreement of the data and theoretical predictions. Aims This paper presents a rigorous derivation of a goodness-of-fit statistics for colour-magnitude diagrams (CMD). We discuss the reliability of the underlying assumptions and their validity in real-world testing. Methods. We derived the distribution of the sum of squared Mahalanobis distances of stellar data and theoretical isochrone for a generic set of data and models. We applied this to the case of synthetic CMDs that were constructed to mimic real data of open clusters in the Gaia sample. Then, we analysed the capability of distinguishing among different sets of input physics and parameters that were used to compute the stellar models. To do this, we generated synthetic clusters from isochrones computed with these perturbed quantities, and we evaluated the goodness-of-fit with respect to the reference unperturbed isochrone. Results. We show that when r magnitudes are available for each of the N observational objects and p hyperparameters are estimated in the fit, the error distribution follows a χ2 distribution with (r − 1)N − p degrees of freedom. We show that the linearisation of the isochrone between support points introduces negligible deviation from this result. We investigated the possibility of detecting the effects on stellar models that are induced when the following physical quantities were varied: convective core overshooting efficiency, 14N(p, γ)15O reaction rate, microscopic diffusion velocities, outer boundary conditions, and colour transformation (bolometric corrections). We conducted the analysis at three different ages, 150 Myr, 1 Gyr, and 7 Gyr, and accounted for errors in photometry from 0.003 mag to 0.03 mag. The results suggest that it is possible to detect the effect induced by only some of the perturbed quantities. The effects induced by a change in the diffusion velocities or in the 14N(p, γ)15O reaction rate are too small to be detected even when the smallest photometric uncertainty is adopted. A variation in the convective core overshooting efficiency was detectable only for photometric errors of 0.003 mag and only for the 1 Gyr case. The effects induced by the outer boundary conditions and the bolometric corrections are the largest; the change in outer boundary conditions is detected for photometric errors below about 0.01 mag, while the variation in bolometric corrections is detectable in the whole photometric error range. As a last exercise, we addressed the validity of the goodness-of-fit statistics for real-world open cluster CMDs, contaminated by field stars or unresolved binaries. We assessed the performance of a data-driven cleaning of observations, aiming to select only single stars in the main sequence from Gaia photometry. This showed that this selection is possible only for a very precise photometry with errors of few millimagnitudes.

Unified gas-kinetic wave-particle methods IV: multi-species gas mixture and plasma transport

In this paper, we extend the unified gas-kinetic wave-particle (UGKWP) methods to the multi-species gas mixture and multiscale plasma transport. The construction of the scheme is based on the direct modeling on the mesh size and time step scales, and the local cell’s Knudsen number determines the flow physics. The proposed scheme has the multiscale and asymptotic complexity diminishing properties. The multiscale property means that according to the cell’s Knudsen number the scheme can capture the non-equilibrium flow physics when the cell size is on the kinetic mean free path scale, and preserve the asymptotic Euler, Navier-Stokes, and magnetohydrodynamics (MHD) when the cell size is on the hydrodynamic scale and is much larger than the particle mean free path. The asymptotic complexity diminishing property means that the total degrees of freedom of the scheme reduce automatically with the decreasing of the cell’s Knudsen number. In the continuum regime, the scheme automatically degenerates from a kinetic solver to a hydrodynamic solver. In the UGKWP, the evolution of microscopic velocity distribution is coupled with the evolution of macroscopic variables, and the particle evolution as well as the macroscopic fluxes is modeled from a time accumulating solution of kinetic scale particle transport and collision up to a time step scale. For plasma transport, the current scheme provides a smooth transition from particle-in-cell (PIC) method in the rarefied regime to the magnetohydrodynamic solver in the continuum regime. In the continuum limit, the cell size and time step of the UGKWP method are not restricted by the particle mean free path and mean collision time. In the highly magnetized regime, the cell size and time step are not restricted by the Debye length and plasma cyclotron period. The multiscale and asymptotic complexity diminishing properties of the scheme are verified by numerical tests in multiple flow regimes.

Spreading Dynamics of a Precursor Film of Ionic Liquid or Water on a Micropatterned Polyelectrolyte Brush Surface.

Time evolution of the microscopic wetting velocity of 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide (EMI-TFSI) or water on a micrometer-scale line-patterned surface with a poly(3-sulfopropyl methacrylate) brush and a hydrophobic perfluoroalkyl monolayer was precisely measured by direct observation using optical microscopy and a selective dyeing method over a long period (178 days). When a liquid droplet was placed on the dyed line-patterned brush surface, the liquid penetrated and spread into the polymer brush layer, forming a precursor thin film that extended beyond the macroscopic contact line. The elongation proceeded in two stages by an adiabatic process followed by a diffusive process. The elongation distance X increased with time in proportion to t2.6 for water and t0.81 for EMI-TFSI during the adiabatic process. In a diffusive process, the advancing velocity of the precursor film was markedly reduced to be expressed as X ∝ t0.66 for water and X ∝ t0.21 for EMI-TFSI, indicating that the diffusive process was affected by the energy dissipation of the wetting system. The high viscosity and the strong molecular interaction of EMI-TFSI with the polymer brush gave a large entropy change during the wetting process to result in a slower spreading velocity.