Large Knudsen Numbers Research Articles

Accuracy and performance evaluation of low density internal and external flow predictions using CFD and DSMC

The Direct Simulation Monte Carlo (DSMC) method was widely used to simulate low density gas flows with large Knudsen numbers. However, DSMC encounters limitations in the regime of lower Knudsen numbers (Kn<0.05). In such cases, approaches from classical computational fluid dynamics (CFD) relying on the continuum assumption are preferred, offering accurate solutions at acceptable computational costs. In experiments aimed at imaging aerosolized nanoparticles in vacuo a wide range of Knudsen numbers occur, which motivated the present study on the analysis of the advantages and drawbacks of DSMC and CFD simulations of rarefied flows in terms of accuracy and computational effort. Furthermore, the potential of hybrid methods is evaluated. For this purpose, DSMC and CFD simulations of the flow inside a convergent–divergent nozzle (internal expanding flow) and the flow around a conical body (external shock generating flow) were carried out. CFD simulations utilize the software OpenFOAM and the DSMC solution is obtained using the software SPARTA. The results of these simulation techniques are evaluated by comparing them with experimental data (1), evaluating the time-to-solution (2) and the energy consumption (3), and assessing the feasibility of hybrid CFD-DSMC approaches (4).

Unified gas-kinetic wave–particle method for polydisperse gas–solid particle multiphase flow

The gas-particle flow with multiple dispersed solid phases is associated with a complicated multiphase flow dynamics. In this paper, a unified algorithm is proposed for the gas-particle multiphase flow. The gas-kinetic scheme (GKS) is used to simulate the gas phase and the multiscale unified gas-kinetic wave–particle (UGKWP) method is developed for the multiple dispersed solid particle phase. For each disperse solid particle phase, the decomposition of deterministic wave and statistic particle in UGKWP is based on the local cell's Knudsen number. The method for solid particle phase can become the Eulerian fluid approach at the small cell's Knudsen number and the Lagrangian particle approach at the large cell's Knudsen number. This becomes an optimized algorithm for simulating dispersed particle phases with a large variation of Knudsen numbers due to different physical properties of the individual particle phase, such as the particle diameter, material density, etc. The GKS-UGKWP method for gas-particle flow unifies the Eulerian–Eulerian and Eulerian–Lagrangian methods. The particle and wave decompositions for the solid particle phase and their coupled evolution in UGKWP come from the consideration to balance the physical accuracy and numerical efficiency. Two cases of a gas–solid fluidization system, i.e. one circulating fluidized bed and one turbulent fluidized bed, are simulated. The typical flow structures of the fluidized particles are captured, and the time-averaged variables of the flow field agree well with the experimental measurements. In addition, the shock particle–bed interaction is studied by the proposed method, which validates the algorithm for the polydisperse gas-particle system in the highly compressible case, where the dynamic evolution process of the particle cloud is investigated.

Nonequilibrium Flow Simulations Using Unified Gas-Kinetic Wave-Particle Method

Nonequilibrium flows are commonly encountered in aerospace engineering applications such as spacecraft reentry, rocket launch, and satellite attitude control. Numerical simulations help greatly in the understanding of nonequilibrium flow, multiscale effects, and the dynamics in these applications. Coupling particle transport and collision within the gas evolution process, the unified gas-kinetic wave-particle (UGKWP) method has been developed for multiscale flow simulation, offering a strategy that strikes a balance between accuracy and efficiency. In the present study, the UGKWP method simulates challenging flow problems with large Knudsen number variations, including supersonic flow around a sphere, hypersonic flow around a space vehicle, nozzle plume into vacuum, and side-jet impingement on the hypersonic flow. The UGKWP accurately captures complex flow structures and has been verified through experimental data or direct simulation Monte Carlo results. Regarding computational cost, the UGKWP method requires only 60 GiB of memory to simulate the three-dimensional space vehicle with 560,593 cells under various flow conditions, which is manageable even on personal workstations. Given its excellent efficiency and accuracy, the UGKWP method shows its substantial benefits in simulating multiscale flow for aerospace engineering applications.

The direct Monte Carlo simulation of microchannel flows for a large Knudsen number range

In recent years, porous materials containing micro- and nano-scale pores have found widespread applications. As the pore size decreases in such materials, rarefaction effects become significant in the pore flow, making the study of flow characteristics under higher Knudsen number conditions particularly crucial. In this work, through a direct simulation Monte Carlo (DSMC) method, an in-depth investigation is conducted into the gas flow characteristics and Klinkenberg effect in porous media with pore sizes ranging from 1 nm to 50 μm and Knudsen numbers spanning from 0.02 (slip flow) to 1200 (free molecular flow). The feasibility of using the DSMC method to simulate an internal free molecular flow in a porous medium under extreme rarefaction conditions with a Knudsen number of 1200 is validated. Furthermore, the impact of the gas pressure and porous medium pore size on the permeability is examined. The results reveal that with an increase in the Knudsen number, the dominant forces in the flow field transition from viscous forces to Knudsen diffusion, leading to a gradual increase in permeability. A comparative analysis reveals that existing apparent permeability models only provide satisfactory descriptions under certain Knudsen number conditions. Re-fitting the coefficient of the Kawagoe model and incorporating viscosity corrections leads to an apparent permeability model that can provide good predictions over a broader range of Knudsen numbers.

Simulation of Rarefied Gas Flow in a Channel Applying Artificial Neuron Network

The purpose of the present paper is to confirm numerically applying our ANN algorithm for transitional gas flow between continuum and free-molecular flow. The main advantage of this approach is substantial computing time economy compared to usual DSMC procedure. The learning of ANN is based on well known rarefied gas flow examples, analytic solutions in limit cases, calculations by lattice Boltzmann method (LBM), and on quite few Monte Carlo simulations. Our previous results obtained for large Knudsen numbers have been applied near the free-molecular regime. In this region the flow has demonstrated the properties of unstable nonlinear dynamic system. The flow of nitrogen in a microchannel with aspect ratio L/h = 20 and by pressure ratio P between 1.84 and 4.14 is simulated by ANN. This example has shown a good coincidence of ANN and DSMC calculations even in the case when only a small part of original DSMC computation has been applied for learning ANN. Thus, computing time is reduced up to 10 times in this case.

Simulation of the Knudsen Pump by means of Quasi Gasdynamic Equation System

The use of miniature micropumps based on the effect of thermal gas sliding along the walls with an applied temperature gradient (Knudsen pump) is very relevant when creating modern miniature devices. Currently, algorithmically complex methods such as DSMC are usually used to simulate gas flows in such pumps. Progress in the development of algorithms based on a quasi-gasdynamic system of equations, pushed the authors to apply such algorithm to modeling the flow of a binary gas mixture gases in one variant of a micropump. The equations of the QGD system are conservation laws with additional dissipative terms. These terms contain a coefficient determined by the characteristic time between particles collisions. In this paper, the problem of flow in a rectangular Knudsen pump is solved based on this algorithm for a gas mixture in a single-fluid approximation. Different variants of the composition of the gas mixture are considered. The Knudsen numbers for which it is possible to describe gas flows by the QGD system correspond to flows of dense or moderately rarefied gases. Calculations for not too large Knudsen numbers demonstrated a good agreement with the results obtained by the DSMC method which is much more complex and expensive.

Implementation of nonlocal non-Fourier heat transfer for semiconductor nanostructures

The study of heat transport in micro/nanoscale structures due to their application, especially in Nanoelectronics, is a matter of interest. In other words, the precise simulation of the temperature distribution inside the transistors is consequential in designing and building more reliable devices reaching lower maximum temperatures during the operation. The present study constitutes a framework for micro/nanoscale heat transport study, which leads to calculating accurate temperature/heat flux profiles with low computational cost. The newly non-dimensional parameter γ, presenting the strength of the nonlocality, is utilized through the nonlocal DPL modeling (NDPL). Alongside calculating the nonlocality coefficient, the factors appearing in DPL, including the temperature jump and phase lagging ratio, are revisited. The factor γ is found to have a linear relationship with Knudsen (Kn) number, being 3.5 for Kn=10 and 0.035 for Kn=0.1. Although the nonlocality is bold for the large Knudsen numbers, it also plays a vital role for low Knudsen number structures, especially at earlier times. Further, It is obtained that intruding γ is critical for obtaining accurate temperature and heat flux distributions that are very close to the practical results of the Phonon Boltzmann equation.

Generalized Reynolds equation for microscale lubrication between eccentric circular cylinders based on kinetic theory

A microscale lubrication flow of a gas between eccentric circular cylinders is studied on the basis of kinetic theory. The dimensionless curvature, defined by the mean clearance divided by the radius of the inner cylinder, is small, and the rotation speed of the inner cylinder is also small. The Knudsen number, defined by the mean free path divided by the mean clearance, is arbitrary. The Boltzmann equation is studied analytically using the slowly varying approximation following the method proposed in the author's previous study (Doi,Phys. Rev. Fluids, vol. 7, 2022, 034201). A macroscopic lubrication equation, which is a microscale generalization of the Reynolds lubrication equation, is derived. To assess this, a direct numerical analysis of the Boltzmann equation in a bipolar coordinate system is conducted using the Bhatnagar–Gross–Krook–Welander kinetic equation. It is demonstrated that the solution of the derived lubrication equation approximates that of the Boltzmann equation over a wide range of the eccentricity and the whole range of the Knudsen number. It is also demonstrated that another lubrication equation derived by a formal application of the slowly varying approximation produces a non-negligible error of the order of the square root of the dimensionless curvature for large Knudsen numbers.

Adaptive wave-particle decomposition in UGKWP method for high-speed flow simulations

With wave-particle decomposition, a unified gas-kinetic wave-particle (UGKWP) method has been developed for multiscale flow simulations. With the variation of the cell Knudsen number, the UGKWP method captures the transport process in all flow regimes without the kinetic solver’s constraint on the numerical mesh size and time step being determined by the kinetic particle mean free path and particle collision time. In the current UGKWP method, the cell Knudsen number, which is defined as the ratio of particle collision time to numerical time step, is used to distribute the components in the wave-particle decomposition. The adaptation of particles in the UGKWP method is mainly for the capturing of the non-equilibrium transport. In this aspect, the cell Knudsen number alone is not enough to identify the non-equilibrium state. For example, in the equilibrium flow regime with a Maxwellian distribution function, even at a large cell Knudsen number, the flow evolution can be still modelled by the Navier-Stokes solver. More specifically, in the near space environment both the hypersonic flow around a space vehicle and the plume flow from a satellite nozzle will encounter a far field rarefied equilibrium flow in a large computational domain. In the background dilute equilibrium region, the large particle collision time and a uniform small numerical time step can result in a large local cell Knudsen number and make the UGKWP method track a huge number of particles for the far field background flow in the original approach. But, in this region the analytical wave representation can be legitimately used in the UGKWP method to capture the nearly equilibrium flow evolution. Therefore, to further improve the efficiency of the UGKWP method for multiscale flow simulations, an adaptive UGKWP (AUGKWP) method is developed with the introduction of an additional local flow variable gradient-dependent Knudsen number. As a result, the wave-particle decomposition in the UGKWP method is determined by both the cell and gradient Knudsen numbers, and the use of particles in the UGKWP method is solely to capture the non-equilibrium flow transport. The current AUGKWP method becomes much more efficient than the previous one with the cell Knudsen number only in the determination of wave-particle composition. Many numerical tests, including Sod shock tube, normal shock structure, hypersonic flow around cylinder, flow around reentry capsule, and an unsteady nozzle plume flow, have been conducted to validate the accuracy and efficiency of the AUGKWP method. Compared with the original UGKWP method, the AUGKWP method achieves the same accuracy, but has advantages in memory reduction and computational efficiency in the simulation for flows with the co-existing of multiple regimes.

Machine learning of hidden variables in multiscale fluid simulation

Solving fluid dynamics equations often requires the use of closure relations that account for missing microphysics. For example, when solving equations related to fluid dynamics for systems with a large Reynolds number, sub-grid effects become important and a turbulence closure is required, and in systems with a large Knudsen number, kinetic effects become important and a kinetic closure is required. By adding an equation governing the growth and transport of the quantity requiring the closure relation, it becomes possible to capture microphysics through the introduction of ‘hidden variables’ that are non-local in space and time. The behavior of the ‘hidden variables’ in response to the fluid conditions can be learned from a higher fidelity or ab-initio model that contains all the microphysics. In our study, a partial differential equation simulator that is end-to-end differentiable is used to train judiciously placed neural networks against ground-truth simulations. We show that this method enables an Euler equation based approach to reproduce non-linear, large Knudsen number plasma physics that can otherwise only be modeled using Boltzmann-like equation simulators such as Vlasov or particle-in-cell modeling.

Numerical simulation of lateral jet interaction with rarefied hypersonic flow over a two-dimensional blunt body

Reaction Control System (RCS) is a direct force control system that successfully adjusts a craft's attitude or orbit using the reaction force created by jet flow. RCS is frequently employed in the management of near-space vehicles due to its properties of fast response time and effective control efficiency. When the near-space vehicle is navigating at high altitude in a low density atmosphere, the Navier–Stokes equation is no longer applicable. The numerical approach utilized in this study is known as the Conserved Discrete Unified Gas Kinetic Scheme, and the governing equation is the Boltzmann equation, which is not constrained by the continuum hypothesis. In velocity space, an unstructured mesh is utilized, which minimizes the amount of discrete velocity points and considerably increases computation efficiency. The numerical results are in good agreement with the direct simulation Monte Carlo code DS2V when modeling large Knudsen number lateral jet flow. The interaction flow field between hypersonic free stream and lateral jet is then simulated at altitudes of 60–90 km using argon as the working gas and a two-dimensional blunt cone with lateral jet as the study object. Under a fixed jet pressure ratio, preliminary research was conducted on the variation of the lateral jet interference flow field characteristics with the freestream Knudsen number and angle of attack. The differences in surface pressure and heat flux caused by jet opening and shutting are compared. Under rarefied atmospheric conditions, the variation of the force/moment amplification coefficient is given. The numerical results show that when the angle of attack is 0°, the separation area in front of the nozzle and a pair of opposite vortices, which are common in the jet interference flow field, gradually disappear with increasing altitude, but the separation vortex reappears when the angle of attack of the freestream is increased. The high-pressure region generated upstream of the nozzle is the primary cause of the extra force/moment. The density of the main flow decreases as altitude increases, various shock wave patterns of the interference flow field gradually dissipate and the force/moment amplification factor changes considerably. The rarefied gas effect has a significant effect on the lateral jet interference flow.

Comparison of different Gaussian quadrature rules for lattice Boltzmann simulations of noncontinuum Couette flows: From the slip to free molecular flow regimes

The lattice Boltzmann (LB) method can be formulated directly from the Boltzmann equation with the Bhatnagar–Gross–Krook assumption. This kinetic origin stimulates wide interest in applying it to simulate flow problems beyond the continuum limit. In this article, such a thought is examined by simulating Couette flows from the slip to free molecular flow regimes using the LB models equipped with different discrete velocity spaces, derived from the half-range Gauss Hermite (HGH), Gauss Legendre (GL), Gauss Kronrod (GK), and Gauss Chebyshev first and second quadrature rules. It is found that the conventional HGH-based LB models well describe noncontinuum Couette flows in the slip and weak transition flow regimes. Nonetheless, they suffer from significant errors with the further increasing Knudsen number, even if a large number of discrete velocities have been employed. Their results contrast with those by the LB models derived from the other Gaussian quadrature rules, which have far better accuracy at large Knudsen numbers. In particular, the GL- and GK-based LB models well capture the velocity fields of Couette flows in the strong transition and free molecular flow regimes. These numerical simulations in this article highlight the importance of velocity discretization for the LB simulations at different Knudsen numbers. They reveal that the LB models based on the Gauss Hermite (GH) quadrature rule are not always the best choice for simulating low-speed bounded flows at moderate and large Knudsen numbers; under strong noncontinuum conditions, those non-GH-based LB models proposed in this article have yielded more accurate results.

Particle tracking velocimetry study of the nonequilibrium characteristics of a fluidized bed

AbstractThe nonequilibrium characteristic of fluidization is closely related to the mesoscale structures. To help uncover this relationship, we performed a particle‐level experiment in a bubbling fluidized bed making particle tracking velocimetry (PTV) measurements. We found a large Knudsen number not only in bubbles or over bubble interfaces, but also in the dense emulsion where various banded structures exist, indicating a pervasive violation of the local equilibrium. Analysis of these banded structures identified stretching, compression, or shearing between particles. Non‐Gaussian velocity distributions were found throughout the bed, including in the dense emulsion with a small Knudsen number. In future work, we expect that the nonequilibrium characteristic will act as a local criterion for the assessment of various models of bubbling fluidized beds.

Modified heat transfer correction function for modeling multiphase condensing flows in transonic regime

Water droplet formation and growth in the nucleation process are crucial aspects of steam condensation in moist air flow, and they are accompanied by intensive heat exchange between phases. This study proposes a mathematical model for predicting heat transfer coefficients in the condensation process, which is applicable for single-phase and multiphase flows. The proposed model based on the correction function forms a continuous model for the low Knudsen number regime, and in the region of large Knudsen numbers, it employs a simplified droplet growth model based on Hertz and Knudsen's approach derived from the kinetic theorem. The proposed model was compared with existing models based on the continuous approach, that is, models derived by Gyarmathy, Fuchs-Sutigin, and Young, as well as models based on the kinetic theorem. Furthermore, the proposed model was implemented into a numerical CFD tool, and numerical studies were conducted for two representative cases of single-species and multispecies flows, that is, steam and humid-air expansion in a converging-diverging nozzle. For steam studies, the proposed model showed a similar convergence with the experiment as Young's model however, its agreement with the experimental results for multispecies condensation flow was better. The proposed model is promising for the study of condensation when there is rapid droplet growth, which occurs in multispecies flows, even if further study is needed to find a correlation between the condensation coefficient and fluid properties.

Velocity discretization for lattice Boltzmann method for noncontinuum bounded gas flows at the micro- and nanoscale

The lattice Boltzmann (LB) method intrinsically links to the Boltzmann equation with the Bhatnagar–Gross–Krook collision operator; however, it has been questioned to be able to simulate noncontinuum bounded gas flows at the micro- and nanoscale, where gas moves at a low speed but has a large Knudsen number. In this article, this point has been verified by simulating Couette flows at large Knudsen numbers (e.g., Kn=10 and Kn=100) through use of the linearized LB models based on the popular half-range Gauss–Hermite quadrature. The underlying cause for the poor accuracy of these conventional models is analyzed in the light of the numerical evaluation of the involved Abramowitz functions. A different thought on velocity discretization is then proposed using the Gauss–Legendre (GL) quadrature. Strikingly, the resulting GL-based LB models have achieved high accuracy in simulating Couette flows, Poiseuille flows, and lid-driven cavity flows in the strong transition and even free molecular flow regimes. The numerical study in this article reveals an essentially distinct but workable way in constructing the LB models for simulating micro- and nanoscale low-speed gas flows with strong noncontinuum effects.

Influence of slip effect on viscous dissipation heat and lubrication characteristics of Gas Journal Bearing: A multiscale analysis

As operating under quite high bearing speed with quite narrow clearance, the slip effect generally inevitable exists and closely relates to the performance of Gas Journal Bearing (GJB). However, singly the conventional macroscopic method may not be capable of fully exploring the influence of slip effect and seize tiny details such as slip velocity and flow vortex. In this article, both the Lattice Boltzmann Method (LBM) and the Reynolds equation method are applied to analyze the distributions of lubricant viscous dissipation heat, velocity, slip velocity, pressure, journal position and film thickness versus different operating parameters at multiscale. It shows that the slip effect weakens the squeezing function on lubricant film, reduces the viscous dissipation heat peaks, consumes the maximum pressure peaks, and decreases the bearing capacity, which is more obvious at larger Knudsen number. At higher bearing speed, the high viscous dissipation heat and pressure zone rapidly magnifies, in contrast, under larger bearing load, the high viscous dissipation heat zone concentrates.

A fast-converging scheme for the phonon Boltzmann equation with dual relaxation times

The phonon Boltzmann transport equation with dual relaxation times is often used to describe the heat conduction in semiconductor materials when the classical Fourier's law is no longer valid. For practical engineering designs, accurate and efficient numerical methods are highly demanded to solve the equation. At a large Knudsen number (i.e., the ratio of the phonon mean free path to a characteristic system length), steady-state solutions can be obtained via the conventional iterative scheme (CIS) within a few iterations. However, when the Knudsen number becomes small, i.e., when the phonon transport occurs in the diffusive or hydrodynamic regime, thousands of iterations are required to obtain converged results. In this work, a general synthetic iterative scheme (GSIS) is proposed to tackle the inefficiency of CIS. The key ingredient of the GSIS is that a set of macroscopic synthetic equations, which is exactly derived from the Boltzmann transport equation, is simultaneously solved with the kinetic equation to obtain the temperature and heat flux. During the iteration, the macroscopic quantities are used to evaluate the equilibria in the scattering terms of the kinetic equation, thus guiding the evolution of the phonon distribution function, while the distribution function, in turn, provides closures to the synthetic equations. The Fourier stability analysis is conducted to reveal the superiority of the GSIS over the CIS in terms of fast convergence in periodic systems. It is shown that the convergence rate of the GSIS can always be maintained under 0.2 so that only two iterations are required to reduce the iterative error by one order of magnitude. Numerical results in wall-bounded systems are presented to demonstrate further the efficiency of GSIS, where the CPU time is reduced by up to three orders of magnitude, especially in both the diffusive and hydrodynamic regimes where the Knudsen number is small.

Effect of a small curvature of the surfaces on microscale lubrication of a gas for large Knudsen numbers

Lubrication flow of a gas in a microscale gap between coaxial circular cylinders with a nonuniform accommodation coefficient is studied on the basis of kinetic theory. The Boltzmann equation is studied analytically using the slowly varying approximation when the dimensionless curvature, namely the gap size divided by the radius of the inner cylinder, is small. It is demonstrated that a nonnegligible effect of small curvature affects the lubrication performance when the Knudsen number is large.

On the Test Particle Monte-Carlo method to solve the steady state Boltzmann equation, the congruity of its results with experiments and its potential for shared memory parallelism

The Test Particle Monte Carlo is a known method to solve the steady state Boltzmann particle transport equation in rarefied gas systems. A description of the Test Particle Monte-Carlo procedure is outlined and the accuracy of this method is investigated by analyzing its consistency to experimental data of steady state effusions in isothermal systems. Computational results present deviations from the experiments that are at most 6.7%. After which, this paper investigates the potential of this method when it is parallelized using multi-core CPUs with shared memory for large Knudsen numbers. Scalability is expected since the method relies on a mean particle field that renders particle trajectories nearly independent from one another, therefore reducing data communication between processing cores. The mean particle distribution field helps linearize the collision term of the Boltzmann equation. Shared Memory Parallelism is an interesting feature when combined with distributed memory parallelism for design optimization of vacuum systems.

Measuring Gas Transport and Sorption in Large Intact Geologic Specimens via the Piezometric Method

Uptake of noble gases into heterogeneous geologic core samples was measured using a piezometric methodology. In addition to measuring accessible porosity—as with gas pycnometry—by monitoring the rate of pressure decay, this method can also be used to estimate the gas effective diffusivity in the sample. In contrast to previous applications of this method, where milligram quantities of fractured grains are characterized, here approximately kilogram core samples were left intact when tested. In doing so, a more representative sample of the heterogeneous field geology is provided. Additionally, alteration of the pore structure and connectivity during sample preparation is avoided. To scale the piezometric method from milligrams to kilograms, the system was designed to operate at medium vacuum (1 to 100 Pa) to restrict transport in pores less than approximately 60 µm to large Knudsen numbers. To test the system performance, two samples of interest were selected: a rhyolitic welded tuff from Blue Canyon Dome at the Energetic Materials Research and Testing Center and a zeolitized non-welded rhyolitic tuff from the Nevada National Security Site. Three noble gases were utilized in this test series; Argon and xenon as they are of direct interest to nuclear monitoring efforts and helium as it is a weakly adsorbing reference standard. Additionally, mercury intrusion porosimetry measurements were made on subsamples of the core to compare the observed porosity by the two methods and to discuss gas transport rates in the context of the measured pore distribution.