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Abstract The kinetic theory of rarefied gases and numerical schemes based on the Boltzmann equation, have evolved to the cornerstone of non-equilibrium gas dynamics. However, their counterparts in the dense regime remain rather exotic for practical non-continuum scenarios. This problem is partly due to the fact that long-range interactions arising from the attractive tail of molecular potentials, lead to a computationally demanding Vlasov integral. This study focuses on numerical remedies for efficient stochastic particle simulations based on the Enskog–Vlasov kinetic equation. In particular, we devise a Poisson type elliptic equation which governs the underlying long-range interactions. The idea comes through fitting a Green function to the molecular potential, and hence deriving an elliptic equation for the associated fundamental solution. Through this transformation of the Vlasov integral, efficient Poisson type solvers can be readily employed in order to compute the mean field forces. Besides the technical aspects of different numerical schemes for treatment of the Vlasov integral, simulation results for evaporation of a liquid slab into the vacuum are presented. It is shown that the proposed formulation leads to accurate predictions with a reasonable computational cost. Highlights • Reviewing a thorough treatment of molecular interactions in the kinetic framework. • Translating the n-body problem underlying the mean-field description of long range interactions to a Poisson type PDE. • Validating the devised solution algorithm for liquids and dense gases in Couette and evaporation scenarios. [ABSTRACT FROM AUTHOR]

In this paper, the influence of the gas-surface interaction on the time-dependent rarefied gas flow through a short tube into vacuum is investigated. Due to a significant scale separation, the flow is simulated using a hybrid scheme proposed by Vargas et al. [1, 2], in which the pressure change in the upstream chamber is coupled to the flow rate obtained by the Direct Simulation Monte Carlo (DSMC) computations. First, the influence of gas-surface interaction is demonstrated through comparison of DSMC simulation results obtained for different values of accommodation coefficients. The obtained dataset is then used to show how the boundary conditions can affect the time-dependent gas flow, even when applied on the small surface area of the device. It is argued that the domain of possible solutions is bounded by the cases of specular and diffuse scattering. Furthermore, it is investigated how this domain can be affected by measurement uncertainties present in the experimental setup. Next, the analysis of the gas-surface interaction is extended to binary mixtures. In the end, it is discussed how the obtained results can be used in order to improve the efficiency of gas separation mechanisms. [ABSTRACT FROM AUTHOR]

Recently, Fokker–Planck based particle Monte Carlo schemes have been proposed and evaluated for simulations of rarefied gas flows [1–3] . In this paper, the variance reduction for particle Monte Carlo simulations based on the Fokker–Planck model is considered. First, deviational based schemes were derived and reviewed, and it is shown that these deviational methods are not appropriate for practical Fokker–Planck based rarefied gas flow simulations. This is due to the fact that the deviational schemes considered in this study lead either to instabilities in the case of two-weight methods or to large statistical errors if the direct sampling method is applied. Motivated by this conclusion, we developed a novel scheme based on correlated stochastic processes. The main idea here is to synthesize an additional stochastic process with a known solution, which is simultaneously solved together with the main one. By correlating the two processes, the statistical errors can dramatically be reduced; especially for low Mach numbers. To assess the methods, homogeneous relaxation, planar Couette and lid-driven cavity flows were considered. For these test cases, it could be demonstrated that variance reduction based on parallel processes is very robust and effective. [ABSTRACT FROM AUTHOR]

A Fokker–Planck based particle Monte Carlo algorithm was devised recently for simulations of rarefied gas flows by the authors [1–3] . The main motivation behind the Fokker–Planck (FP) model is computational efficiency, which could be gained due to the fact that the resulting stochastic processes are continuous in velocity space. This property of the model leads to simulations where the computational cost becomes independent of the Knudsen number (Kn) [3] . However, the Fokker–Planck model which can be seen as a diffusion approximation of the Boltzmann equation, becomes less accurate as Kn increases. In this study we propose a hybrid Fokker–Planck–Direct Simulation Monte Carlo (FP–DSMC) solution method, which is applicable for the whole range of Kn. The objective of this algorithm is to retain the efficiency of the FP scheme at low Kn ( Kn ≪ 1 ) and to employ conventional DSMC at high Kn ( Kn ≫ 1 ). Since the computational particles employed by the FP model represent the same data as in DSMC, the coupling between the two methods is straightforward. The new ingredient is a switching criterion which would ideally result in a hybrid scheme with the efficiency of the FP method and the accuracy of DSMC for the whole Kn-range. Here, we adopt the number of collisions in a given computational cell and for a given time step size as a decision criterion in order to switch between the FP model and DSMC. For assessment of the hybrid algorithm, different test cases including flow impingement and flow expansion through a slit were studied. Both accuracy and efficiency of the model are shown to be excellent for the presented test cases. [ABSTRACT FROM AUTHOR]

This work presents a kinetic wall boundary model for diatomic gas molecules. The model is derived by generalizing the Cercignani-Lampis-Lord gas-surface interaction kernel in order to account for the gas internal degrees of freedom. Here, opposed to the extensions by Lord ["Some extensions to the Cercignani-Lampis gas-surface scattering kernel," Phys. Fluids 3, 706-710 (1991)], energy exchange between different molecular modes is honored and thus, different physical phenomena arising from inelastic gas-surface collisions can be described. For practical implementations of the model, a Monte-Carlo algorithm was devised, which significantly reduces the computational cost associated with sampling. Comparisons of model predictions with experimental and molecular dynamics data exhibit good agreement. Moreover, simulation studies are performed to demonstrate how energy transfers between different modes due to wall collisions can be exploited for gas separation. [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to the algorithmic improvement and careful analysis of the Fokker–Planck kinetic model derived by Jenny et al. [1] and Gorji et al. [2]. The motivation behind the Fokker–Planck based particle methods is to gain efficiency in low Knudsen rarefied gas flow simulations, where conventional direct simulation Monte Carlo (DSMC) becomes expensive. This can be achieved due to the fact that the resulting model equations are continuous stochastic differential equations in velocity space. Accordingly, the computational particles evolve along independent stochastic paths and thus no collision needs to be calculated. Therefore the computational cost of the solution algorithm becomes independent of the Knudsen number. In the present study, different computational improvements were persuaded in order to augment the method, including an accurate time integration scheme, local time stepping and noise reduction. For assessment of the performance, gas flow around a cylinder and lid driven cavity flow were studied. Convergence rates, accuracy and computational costs were compared with respect to DSMC for a range of Knudsen numbers (from hydrodynamic regime up to above one). In all the considered cases, the model together with the proposed scheme give rise to very efficient yet accurate solution algorithms. [Copyright &y& Elsevier]

A Fokker-Planck based kinetic model is presented here, which also accounts for internal energy modes characteristic for diatomic gas molecules. The model is based on a Fokker-Planck approximation of the Boltzmann equation for monatomic molecules, whereas phenomenological principles were employed for the derivation. It is shown that the model honors the equipartition theorem in equilibrium and fulfills the Landau-Teller relaxation equations for internal degrees of freedom. The objective behind this approximate kinetic model is accuracy at reasonably low computational cost. This can be achieved due to the fact that the resulting stochastic differential equations are continuous in time; therefore, no collisions between the simulated particles have to be calculated. Besides, because of the devised energy conserving time integration scheme, it is not required to resolve the collisional scales, i.e., the mean collision time and the mean free path of molecules. This, of course, gives rise to much more efficient simulations with respect to other particle methods, especially the conventional direct simulation Monte Carlo (DSMC), for small and moderate Knudsen numbers. To examine the new approach, first the computational cost of the model was compared with respect to DSMC, where significant speed up could be obtained for small Knudsen numbers. Second, the structure of a high Mach shock (in nitrogen) was studied, and the good performance of the model for such out of equilibrium conditions could be demonstrated. At last, a hypersonic flow of nitrogen over a wedge was studied, where good agreement with respect to DSMC (with level to level transition model) for vibrational and translational temperatures is shown. [ABSTRACT FROM AUTHOR]

We introduce a framework for generating samples of a distribution given a finite number of its moments, targeted at particle-based solutions of kinetic equations and rarefied gas flow simulations. Our model, referred to as the Wasserstein-Entropy distribution (WE), couples a physically-motivated Wasserstein penalty term to the traditional maximum-entropy distribution (MED) function, which serves to regularize the latter. The penalty term becomes negligible near the local equilibrium, reducing the proposed model to the MED, known to reproduce the hydrodynamic limit. However, in contrast to the standard MED, the proposed WE closure can cover the entire physically realizable moment space, including the so-called Junk line. We also propose an efficient Monte Carlo algorithm for generating samples of the unknown distribution which is expected to outperform traditional non-linear optimization approaches used to solve the MED problem. Numerical tests demonstrate that given moments up to the heat flux—that is, information equivalent to that contained in the Chapman-Enskog distribution—the proposed methodology provides a reliable closure in the collision-dominated and early transition regimes. Applications to greater rarefaction demand information from higher-order moments, which can be incorporated within the proposed closure. • Generate samples from distribution defined by small number of its moments. • Solution based on entropic optimal transport. • Resulting distribution is well defined for all realizable moments. • Samples generated using efficient Monte Carlo algorithm. • Validated using DSMC simulations of rarefied gas hydrodynamics. [ABSTRACT FROM AUTHOR]

• Devising a class of Fokker-Planck equations consistent with H-theorem. • Devising a kinetic model for arbitrary molecular potential. • Presenting efficient particle solution algorithm for rarefied gas flows. • Testing the devised solution algorithm in high Mach/low speed flows. The diffusion limit of kinetic systems has been subject of numerous studies since prominent works of Lebowitz et al. [1] and van Kampen [2]. More recently, the topic has seen a fresh interest from the rarefied gas simulation perspective. In particular, Fokker-Planck based kinetic models provide novel approximations of the Boltzmann equation, where the relaxation induced by binary collisions is modeled via continuous stochastic processes. Hence in contrast to direct simulation Monte-Carlo, computational particles follow seemingly independent stochastic paths. As a result, a significant computational gain at small/vanishing Knudsen numbers can be obtained, where the dynamics of particles is overwhelmed by collisions. The cubic Fokker-Planck equation derived by [3] gives rise to the correct viscosity and Prandtl number for monatomic gases in the hydrodynamic limit, and further accurate behavior at moderate Knudsen numbers. Yet the model lacks a rigorous structure and more crucially does not admit the H-theorem. The latter underpins its accuracy e.g. in predicting shock wave profiles. This study addresses bridging the gap between diffusion processes and the Boltzmann equation by introducing the Entropic-Fokker-Planck kinetic model. The drift-diffusion closures derived for the model, allow for an H-theorem besides honoring consistent relaxation of moments. The devised model is validated with respect to direct simulation Monte-Carlo for high-Mach as well as Couette flows. Good performance of the model together with its easy to compute coefficients, makes the Entropic-Fokker-Planck framework attractive for computational investigation of gases beyond equilibrium. [ABSTRACT FROM AUTHOR]

• Applying machine learning (ANN/GP) on closure problem by maximum entropy ansatz. • Recovering bi-modal density using data-driven maximum entropy distribution. • Leveraging machine learning for hybrid kinetic-continuum particle solver. • Validating data-driven NSF-Boltzmann solver for Sod's shock tube. • Achieving accurate, efficient, and robust hybrid particle solver. An important class of multi-scale flow scenarios deals with an interplay between kinetic and continuum phenomena. While hybrid solvers provide a natural way to cope with these settings, two issues restrict their performance. Foremost, the inverse problem implied by estimating distributions has to be addressed, to provide boundary conditions for the kinetic solver. The next issue comes from defining a robust yet accurate switching criterion between the two solvers. This study introduces a data-driven kinetic-continuum coupling, where the Maximum-Entropy-Distribution (MED) is employed to parametrize distributions arising from continuum field variables. Two regression methodologies of Gaussian-Processes (GPs) and Artificial-Neural-Networks (ANNs) are utilized to predict MEDs efficiently. Hence the MED estimates are employed to carry out the coupling, besides providing a switching criterion. To achieve the latter, a continuum breakdown parameter is defined by means of the Fisher information distance computed from the MED estimates. We test the performance of our devised MED estimators by recovering bi-modal densities. Next, MED estimates are integrated into a hybrid kinetic-continuum solution algorithm. Here Direct Simulation Monte-Carlo (DSMC) and Smoothed-Particle Hydrodynamics (SPH) are chosen as kinetic and continuum solvers, respectively. The problem of monatomic gas inside Sod's shock tube is investigated, where DSMC-SPH coupling is realized by applying the devised MED estimates. Very good agreements with respect to benchmark solutions along with a promising speed-up are observed in our reported test cases. [ABSTRACT FROM AUTHOR]

• Devising Bayesian inference for fast evaluation of maximum entropy distribution. • Adopting Radial basis function for covariance kernel. • Excellent recovery of bi-modal densities among others. Maximum-Entropy Distributions offer an attractive family of probability densities suitable for moment closure problems. Yet finding the Lagrange multipliers which parametrize these distributions, turns out to be a computational bottleneck for practical closure settings. Motivated by recent success of Gaussian processes, we investigate the suitability of Gaussian priors to approximate the Lagrange multipliers as a map of a given set of moments. Examining various kernel functions, the hyperparameters are optimized by maximizing the log-likelihood. The performance of the devised data-driven Maximum-Entropy closure is studied for couple of test cases including relaxation of non-equilibrium distributions governed by Bhatnagar-Gross-Krook and Boltzmann kinetic equations. [ABSTRACT FROM AUTHOR]

Motivated by improving the performance of particle-based Monte-Carlo simulations in the transitional regime, Fokker-Planck kinetic models have been devised and studied as approximations of the Boltzmann collision operator. By generalizing the linear drift model, the cubic Fokker-Planck (cubic-FP) and ellipsoidal Fokker-Planck (ES-FP) have been proposed, in order to obtain the correct Prandtl number of 2/3 for a dilute monatomic gas. This study provides a close comparison between both models in low Mach and supersonic settings. While direct simulation Monte-Carlo (DSMC) here serves as the benchmark, overall close performance between cubic-FP, ES-FP, and DSMC is observed. Furthermore, while the ES-FP outperforms the cubic-FP model in the shock region of the supersonic flow around a cylinder, the latter shows a better accuracy in the near continuum regime. It is argued that the reason behind these discrepancies lies in the entropy law besides the transport properties. [ABSTRACT FROM AUTHOR]