# A solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion

DOI: 10.1016/j.jcp.2009.10.008 · Source: DBLP

Abstract

In this paper, a stochastic model is presented to simulate the flow of gases, which are not in thermodynamic equilibrium, like in rarefied or micro situations. For the interaction of a particle with others, statistical moments of the local ensemble have to be evaluated, but unlike in molecular dynamics simulations or DSMC, no collisions between computational particles are considered. In addition, a novel integration technique allows for time steps independent of the stochastic time scale.The stochastic model represents a Fokker–Planck equation in the kinetic description, which can be viewed as an approximation to the Boltzmann equation. This allows for a rigorous investigation of the relation between the new model and classical fluid and kinetic equations. The fluid dynamic equations of Navier–Stokes and Fourier are fully recovered for small relaxation times, while for larger values the new model extents into the kinetic regime.Numerical studies demonstrate that the stochastic model is consistent with Navier–Stokes in that limit, but also that the results become significantly different, if the conditions for equilibrium are invalid. The application to the Knudsen paradox demonstrates the correctness and relevance of this development, and comparisons with existing kinetic equations and standard solution algorithms reveal its advantages. Moreover, results of a test case with geometrically complex boundaries are presented.

- ... This is due to the fact that the collision rate scales with the density squared which leads to a significant number of jumps implied by the Markov process underlying the Enskog collision operator. As an efficient alternative, following the methodology proposed by Jenny et al. [29] and Gorji et al. [30,31] for ideal gases, a Fokker-Planck model as an approximation of the Enskog operator was introduced by the authors [32]. In the Fokker-Planck approach, the effect of collisions are modeled through drift and diffusion actions which lead to continuous stochastic paths instead of binary collisions. ...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.
- ... An alternative stochastic particle scheme, termed Fokker-Planck (FP) method, approximates the collision process underlying the Boltzmann equation by a FP kinetic model [14,15] . In the cubic FP approach, the collision integral of the Boltzmann equation is mod- eled by a drift and diffusion stochastic process, where the parti- cles follow independent stochastic paths. ...... based on the Langevin equation [14] . Time scale τ = 2 μ/p is em- ployed in the cubic FP model which is proportional to the mean collision time of Maxwell molecules. ...... The coefficients of the flow through an orifice arise from interpolating the data obtained in this work. It is known that the cubic FP scheme can safely be used for Knudsen numbers up to one [14][15][16] . However, it is not clear that the cubic FP scheme provides an accurate solu- tion at highly rarefied flow. ...Article
- Aug 2018
- COMPUT FLUIDS

The flow through a thin slit and a thin orifice is studied with the Direct Simulation Monte Carlo (DSMC), cubic Fokker–Planck (FP), and a coupled FP-DSMC hybrid method. Pressure driven monatomic gas flows through a slit and an orifice with various values of degree of rarefaction and pressure ratio are computed. The DSMC method is physically accurate for all flow regimes; however it is computationally expensive in high density, near continuum regions. An alternative stochastic particle scheme, the cubic FP kinetic model has addressed this issue by approximating the particle collisions involved in the Boltzmann collision integral with continuous stochastic processes. The ability of the cubic FP method to reproduce breakdown of translational equilibrium is discussed. In addition, a coupled FP-DSMC hybrid scheme is employed aiming at an efficient and accurate solution. The FP-DSMC hybrid scheme employs DSMC in rarefied regions and FP method in near continuum flow regions. Numerical procedures of the cubic FP method are implemented within the framework of an existing DSMC-solver, SPARTA. The FP-DSMC hybrid solution reproduces pure DSMC solution with improved computational efficiency up to a factor of eight for vacuum flow through a thin slit. - ... Since the seminal work of Jenny etc. [34], the stochastic particle method based on the Fokker-Planck model has been developed and applied widely [35][36][37][38][39][40]. The integral solution of the Fokker-Planck model naturally couples the molecular convection and collision, and hence theoretically its viscosity and thermal conductivity can satisfy the NS solutions at large time steps [37,40]. ...... For steady flows, an exponentially weighted time averaging method [34] is used to reduce statistical noise in sampling. Specifically, the macro variables Q is calculated as ...PreprintFull-text available
- Aug 2018

The stochastic particle method based on Bhatnagar-Gross-Krook (BGK) or ellipsoidal statistical BGK (ESBGK) model approximates the pairwise collisions in the Boltzmann equation using a relaxation process. Therefore, it is more efficient to simulate gas flows at small Knudsen numbers than the counterparts based on the original Boltzmann equation, such as the Direct Simulation Monte Carlo (DSMC) method. However, the traditional stochastic particle BGK method decouples the molecular motions and collisions in analogy to the DSMC method, and hence its transport properties deviate from physical values as the time step increases. This defect significantly affects its computational accuracy and efficiency for the simulation of multiscale flows, especially when the transport processes in the continuum regime is important. In the present paper, we propose a unified stochastic particle ESBGK (USP-ESBGK) method by combining the molecular convection and collision effects. In the continuum regime, the proposed method can be applied using large temporal-spatial discretization and approaches to the Navier-Stokes solutions accurately. Furthermore, it is capable to simulate both the small scale non-equilibrium flows and large scale continuum flows within a unified framework efficiently and accurately. The applications of USP-ESBGK method to a variety of benchmark problems, including Couette flow, thermal Couette flow, Poiseuille flow, Sod tube flow, cavity flow, and flow through a slit, demonstrated that it is a promising tool to simulate multiscale gas flows ranging from rarefied to continuum regime. - ... In recent years, the Fokker-Planck-type kinetic models [19][20][21][22][23][24][25][26][27][28] attract increasing attention, because they are favorable to the design of an efficient particle algorithm for gas flow simulations. Historically, the Fokker-Planck-type models for fluids have been studied in the past decades by Kirkwood [29] , Lebowitz et al. [30] , and Heinz [31] , just to name a few. ...... Jenny, Torrilhon and Heinz [21] proposed a particle Monte Carlo method based on a FP model which has a scalar diffusion coeffi- cient and a linear drift coefficient, but this original FP model leads to an incorrect Prandtl number of 3/2 for monatomic gases. Next, for the purpose of Prandtl number correction, Gorji et al. intro- duced a cubic-FP model [22] in which the drift coefficient is a cu- bic polynomial of the molecular velocity, and extended the par- ticle algorithm [22,35] accordingly. ...... In this paper, we focus on an ellipsoidal statistical Fokker- Planck (ES-FP) model which is firstly proposed by Mathiaud and Mieussens in 2016 [27] . The ES-FP equation enables the Prandtl number correction through a simple modification to the origi- nal FP equation [21] , where the liner drift coefficient keeps un- changed but the scalar diffusion coefficient is replaced by an anisotropic tensor in analogy with the development from the orig- inal BGK equation [12] to the ES-BGK equation [46] . The correction of Prandtl number in Fokker-Planck modeling is an issue of con- cern in recent years. ...Article
- Apr 2018
- COMPUT FLUIDS

In order to improve the efficiency of particle simulation method for gas flows spanning the rarefied and continuum regimes, one promising approach is to employ the Fokker–Planck-type gas kinetic model. The ellipsoidal statistical Fokker–Planck (ES-FP) model treats the evolution of individual molecular velocity as a continuous stochastic process, which can be equivalently described by the Langevin dynamics where the forces acting on each gas molecule are a linear drag force and an anisotropic fluctuating force. By utilizing the exact stochastic integral solution of the corresponding Langevin equation, the ES-FP model is numerically implemented in a particle Monte Carlo manner. In a validation of the particle scheme, excellent agreement is obtained between the simulation results and the analytical solution for the same initial-value problem of ES-FP equation. The ES-FP particle method is used to simulate the relaxation process in a nonequilibrium gas, and agrees well with the direct simulation Monte Carlo (DSMC) method in the predictions of viscous stress and heat flux. Furthermore, extensive ES-FP particle simulations are performed for Couette flows at different Knudsen numbers ranging from 0.001 to 10, as well as supersonic flat-plate flows at different Knudsen numbers ranging from 0.001 to 0.1. Both flow fields and surface quantities are investigated and compared with the DSMC or theoretical solutions. In the ES-FP simulations, the Prandtl number of gas can be corrected, and the physical behaviors in different flow regimes can be properly captured. With reasonable agreement between the ES-FP and DSMC results, the ES-FP simulation is found to be more efficient than DSMC and can significantly save memory cost and CPU time, especially for multi-dimensional flows in the low-Knudsen-number regime. - ... Recently, an alternative stochastic particle scheme was pro- posed by Jenny et al. and Gorji et al., where the collision process underlying the Boltzmann equation is approximated by a Fokker- Planck (FP) kinetic model [18][19][20] . In the cubic FP approach, the Boltzmann equation is reduced to a drift and diffusion stochastic process, where the particles follow independent stochastic paths. ...... based on the Langevin equation [18] . Time scale τ = 2 μ/p is em- ployed in the cubic FP model which is proportional to the mean collision time of Maxwell molecules. ...... Yet, due to the linear position update give in Eq. (8) , large errors in position statistics can be encountered at very small Knudsen numbers. This problem can be coped by us- ing more accurate position update schemes e.g. the one proposed Jenny et al. [18] . However, for the current study, the Euler posi- tion update is employed due to its simplicity and similarity to the DSMC algorithm. ...Article
- Mar 2018
- COMPUT FLUIDS

Hypersonic vehicles experience a wide range of Knudsen number regimes due to changes in atmospheric density. The Direct Simulation Monte Carlo (DSMC) method is physically accurate for all flow regimes, however it is relatively computationally expensive in high density, and low Knudsen number regions. Recent advances in the Fokker–Planck (FP) kinetic models have addressed this issue by approximating the particle collisions involved in the Boltzmann collision integral with continuous stochastic processes. Furthermore, a coupled FP–DSMC solution method has been devised aiming at a universally efficient yet accurate solution algorithm for rarefied gas flows. Well known Lofthouse case of a generic hypersonic flow about a cylinder (Mach 10, Kn 0.01, Argon) is selected to investigate the performance of a hybrid FP–DSMC implementation. The effect of molecular potential on the accuracy of the scheme is mainly analyzed. Furthermore, spatial resolution of cubic FP scheme is studied. Finally, detailed study of accuracy and efficiency of FP–DSMC hybrid scheme is discussed. It is found that the presented adaptive grid together with the FP–DSMC method results in a factor of six speed up for considered hypersonic flow about a cylinder. - ... Its computational cost, however, becomes prohibitively large in the near continuum range (where Kn is small), since evolution of a particle system in the phase space is simulated by explicit treatment of binary collisions. This problem can be addressed by seamlessly coupling DSMC and the Fokker-Planck (FP) based particle Monte Carlo scheme introduced by Jenny et al. [2,3] to form the FP-DSMC method [4]. FP-based simulations are efficient for low to moderate Kn flows, but become inaccurate for very large Kn, since it is based on the integration of continuous stochastic processes. ...... The Fokker-Planck algorithm reads as follows [2][3][4][5]: Let l ≡ (M l , ω l , ξ l ) T denote the state vector of the particle with index l, where ω ∈ R 2 , ξ ∈ R 2 and M ∈ R 3 denote the vectors of rotational, vibrational and translational degrees of freedom (DoF), respectively. Let the notation ( · ) denote averages over a relevant particle ensemble, e.g. ...... The Fokker-Planck model has been shown to produce accurate results up to moderate Kn [2,3,8]. It is important to note that the simulation procedure itself is very similar to DSMC and differs only in the velocity update step, that is, it uses a different collision operator. ...Article
- Mar 2018
- J COMPUT PHYS

Recently, a parallel Fokker–Planck-DSMC algorithm for rarefied gas flow simulation in complex domains at all Knudsen numbers was developed by the authors. Fokker–Planck-DSMC (FP-DSMC) is an augmentation of the classical DSMC algorithm, which mitigates the near-continuum deficiencies in terms of computational cost of pure DSMC. At each time step, based on a local Knudsen number criterion, the discrete DSMC collision operator is dynamically switched to the Fokker–Planck operator, which is based on the integration of continuous stochastic processes in time, and has fixed computational cost per particle, rather than per collision. In this contribution, we present an extension of the previous implementation with automatic local mesh refinement and parallel load-balancing. In particular, we show how the properties of discrete approximations to space-filling curves enable an efficient implementation. Exemplary numerical studies highlight the capabilities of the new code. - Article
- Feb 2018
- APPL ANAL

The ellipsoidal-Fokker–Planck (ES-FP) model is proposed recently to obtain the correct Prandtl number in the numerical simulation of rarefied gas flows, which also satisfies the same properties as the Boltzmann equation, such as the H-theorem and the conservation laws of the mass, momentum, and energy. In this paper, we consider the Cauchy problem for this ES-FP equation in , and prove the global existence of the unique solution and establish its algebraic time convergence rate to the equilibrium state. - ... The Eulerian part serves to conserve momentum of the particle-based system, which is usually described by the LEs. A similar approach can also be applied to model rarefied gases [30] . It appears that the application range of DPD may be extended to such situations, provided that the small-scale behavior of DPD is fully understood. ...... An LE-based model was applied to rarefied gas flows in Refs. [30] and [45]. The Knudsen number is defined as the ratio of the mean free path to the length scale of interest n K := l 0 /λ. ...... For high Knudsen numbers, length scales of interest are comparable to the mean free path of gas molecules. In the proposed model [30,45] , Lagrangian particles are governed by a modified version of LE and the length scales of interest are comparable with the mean free path of Lagrangian particles. In Fig. 3(a), we demonstrate the similarity of the LE and DPD on the N -particle subregime. ...Article
- Jan 2018
- APPL MATH MECH-ENGL

We investigate the behavior of dissipative particle dynamics (DPD) within different scaling regimes by numerical simulations. The paper extends earlier analytical findings of Ripoll, M., Ernst, M. H., and Espa˜nol, P. (Large scale and mesoscopic hydrodynamics for dissipative particle dynamics. Journal of Chemical Physics, 115(15), 7271–7281 (2001)) by evaluation of numerical data for the particle and collective scaling regimes and the four different subregimes. DPD simulations are performed for a range of dynamic overlapping parameters. Based on analyses of the current auto-correlation functions (CACFs), we demonstrate that within the particle regime at scales smaller than its force cut-off radius, DPD follows Langevin dynamics. For the collective regime, we show that the small-scale behavior of DPD differs from Langevin dynamics. For the wavenumber-dependent effective shear viscosity, universal scaling regimes are observed in the microscopic and mesoscopic wavenumber ranges over the considered range of dynamic overlapping parameters. - While accurate simulations of dense gas flows far from the equilibrium can be achieved by direct simulation adapted to the Enskog equation, the significant computational demand required for collisions appears as a major constraint. In order to cope with that, an efficient yet accurate solution algorithm based on the Fokker-Planck approximation of the Enskog equation is devised in this paper; the approximation is very much associated with the Fokker-Planck model derived from the Boltzmann equation by Jenny et al. [“A solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion,” J. Comput. Phys. 229, 1077–1098 (2010)] and Gorji et al. [“Fokker–Planck model for computational studies of monatomic rarefied gas flows,” J. Fluid Mech. 680, 574–601 (2011)]. The idea behind these Fokker-Planck descriptions is to project the dynamics of discrete collisions implied by the molecular encounters into a set of continuous Markovian processes subject to the drift and diffusion. Thereby, the evolution of particles representing the governing stochastic process becomes independent from each other and thus very efficient numerical schemes can be constructed. By close inspection of the Enskog operator, it is observed that the dense gas effects contribute further to the advection of molecular quantities. That motivates a modelling approach where the dense gas corrections can be cast in the extra advection of particles. Therefore, the corresponding Fokker-Planck approximation is derived such that the evolution in the physical space accounts for the dense effects present in the pressure, stress tensor, and heat fluxes. Hence the consistency between the devised Fokker-Planck approximation and the Enskog operator is shown for the velocity moments up to the heat fluxes. For validation studies, a homogeneous gas inside a box besides Fourier, Couette, and lid-driven cavity flow setups is considered. The results based on the Fokker-Planck model are compared with respect to benchmark simulations, where good agreement is found for the flow field along with the transport properties.
- ... In this context, the Langevin dynamics described by Eq. (2) can be considered as a particle Fokker-Planck model. Originally, Jenny et al. [16] proposed a stochastic particle Fokker-Planck algorithm based on Eq. (2) for simulations of rarefied gas flows. Afterwards, Gorji and Jenny [17] developed a more efficient algorithm. ...... where λ is the molecular mean free path, and ¯ c = √ 2R s T is the most probable molecular thermal speed. Previous studies have demonstrated that the linear Langevin model predicts the Prandtl number (Pr) of gas as 3/2 [16,24], and hence the thermal diffusivity coefficient for our model is ...... To obtain the instantaneous local relaxation time, we first determine the local viscosity coefficient according to the calculated local temperature by using the characteristic of the hard-sphere model, i.e., ν/ν ref = (T /T ref ) 1/2 [23], with ν ref = 1.19 m 2 s −1 at T ref = 273 K for the argon gas model used here. We then determine the local relaxation time using the property of the Langevin model, i.e., τ = 2ν/R s T [16,24]. Having the local temperature, macroscopic velocity, and relaxation time determined, the Langevin equation as Eq. ...It is well known that Brownian motion can be described using Langevin equation. In this paper we extend the application of the Langevin equation to the Rayleigh-Bénard (RB) flow, assuming that each molecule in the system is a Brownian particle colliding with its surrounding molecules. The phenomenon of thermal instability, changing from a conductive to a convective state, is well reproduced by Langevin dynamics simulations. The roles of the drag force and the random force terms in the Langevin equation in triggering thermal instability are elucidated via numerical tests. Furthermore, we demonstrate that the strength of the fluctuation correlations increases as the Rayleigh number approaches the critical value, and the characteristics of the fluctuation correlations below the onset of instability foreshadow the form of the convective patterns emerging above the critical point. The Langevin equation, together with the form of the fluctuation correlations, sheds new light on the mechanism of the RB instability.

- Article
- Jan 2004
- APPL MECH REV

1. Introduction.- 2. Stochastic variables.- 3. Stochastic processes.- 4. The equations of fluid and thermodynamics.- 5. Stochastic models for large-scale turbulence.- 6. Stochastic models for small-scale turbulence.- 7. The unification of turbulence models.- 8. References.- 9. Author index.- 10. Subject index. - BookFull-text available
- Jan 2005

http://www.springer.com/engineering/mechanical+engineering/book/978-3-540-24542-1 - Article
- Nov 1953
- J CHEM PHYS

The theory of the coefficients of shear and bulk viscosity of liquids developed in the third article of this series is applied to the calculation of the coefficients of viscosity of liquid argon at its normal boiling point. The theory of the bulk viscosity, including a previously omitted term due to the rate of dilatation, is presented. With the use of the Lennard‐Jones potential, a radial distribution function which is a much better approximation than the previously used one, and a new approximation to the friction constant, values are obtained for the coefficients of viscosity. - ArticleFull-text available
- Jul 2009

An adaptively coupled continuum-molecular approach for compressible viscous flows in transient calculations is presented. The continuum domain is described by the unsteady compressible Navier-Stokes equations, and the molecular domain is solved by direct simulation Monte Carlo. A strategy is described to extend the overlapped Schwarz method with Dirichlet-Dirichlet boundary-condition coupling procedure to transient simulations. The method has been successfully validated against full direct simulation Monte Carlo results for transient simulations of a one-dimensional shock tube and a two-dimensional pressure-driven slit flow. A sensitivity analysis showed that the used overlapped Schwarz coupling method with Dirichlet-Dirichlet boundary conditions is only weakly sensitive to various parameters (e.g., the exact position of the continuum/molecular interface, the size of the overlap region, and the scatter in the molecular solution), which is a clear advantage over the more commonly used flux-based coupling technique. Another advantage of the chosen coupling method is that the continuum and molecular time steps can be decoupled and that, in general, a coupling time step can be used that is much larger than the molecular time step. The study performed also highlighted a limit of the method: it was found to be necessary to keep the Courant number (based on the coupling time step, the cell size in the continuum region, and the molecular most-probable velocity) below 1 to avoid instabilities. © 2009 by the American Institute of Aeronautics and Astronautics, Inc. - Article
- Jun 1950
- J CHEM PHYS

Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies - Article
- Sep 1997
- PHYS FLUIDS

A particle method applying the probability density function (PDF) approach to turbulent compressible flows is presented. The method is applied to several turbulent flows, including the compressible mixing layer, and good agreement is obtained with experimental data. The PDF equation is solved using a Lagrangian/Monte Carlo method. To accurately account for the effects of compressibility on the flow, the velocity PDF formulation is extended to include thermodynamic variables such as the pressure and the internal energy. The mean pressure, the determination of which has been the object of active research over the last few years, is obtained directly from the particle properties. It is therefore not necessary to link the PDF solver with a finite-volume type solver. The stochastic differential equations (SDE) which model the evolution of particle properties are based on existing second-order closures for compressible turbulence, limited in application to low turbulent Mach number flows. Tests are conducted in decaying isotropic turbulence to compare the performances of the PDF method with the Reynolds-stress closures from which it is derived, and in homogeneous shear flows, at which stage comparison with direct numerical simulation (DNS) data is conducted. The model is then applied to the plane compressible mixing layer, reproducing the well-known decrease in the spreading rate with increasing compressibility. It must be emphasized that the goal of this paper is not as much to assess the performance of models of compressibility effects, as it is to present an innovative and consistent PDF formulation designed for turbulent inhomogeneous compressible flows, with the aim of extending it further to deal with supersonic reacting flows. - Article
- Jul 1997

Calculations are reported for recirculating swirling reacting flows using a joint velocity-scalar probability density function (PDF) method. The PDF method offers significant advantages over conventional finite volume, Reynolds-average-based methods, especially for the computation of turbulent reacting flows. The PDF calculations reported here are based on a newly developed solution algorithm for elliptic flows, and on newly developed models for turbulent frequency and velocity that are simpler than those used in previously reported PDF calculations. Calculations are performed for two different gas-turbine-like swirl combustor flows for which detailed measurements are available. The computed results are in good agreement with experimental data.