A continuous stochastic model for non-equilibrium dense gases

Abstract

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.

Treatment of long-range interactions arising in the Enskog–Vlasov description of dense fluids

“…In the framework of stochastic particle methods, the Enskog operator can be numerically approximated via Consistent Boltzmann Algorithm (CBA) [28], Enskog Simulation Monte-Carlo (ESMC) [24][25][26], Frezzotti's algorithm [27] or the dense gas Fokker-Planck model (DFP) [32]. In the following section, numerical challenges associated with the Vlasov integral are investigated and a novel numerical scheme for an efficient computation of long-range interactions is introduced.…”

“…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.…”

“…Even though the computational complexity resulting from the attractive forces does not scale with the density, it can become the bottle-neck of multi-phase flow simulations once the collision operator is replaced by an efficient continuous stochastic model, such as the Fokker-Planck model [32]. The main objective of this study is to introduce a numerical scheme for solving the Vlasov integral, where the resulting computational complexity is less demanding than the direct method.…”

Lattice Boltzmann simulation of double-diffusive natural convection of viscoplastic fluids in a porous cavity

“…The lattice Boltzmann method (LBM) has been known as a powerful mesoscopic numerical method to model a broad variety of complex fluid flow phenomena. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] Huilgol and Kefayati 16 introduced an innovative LBM that can be applied to all fluids, whether they be Newtonian, or power law fluids, or viscoelastic and viscoplastic fluids. In fact, in this method, the equilibrium distribution function is altered in different directions and nodes while the relaxation time is fixed.…”

Particle number control for direct simulation Monte-Carlo methodology using kernel estimates

“…5,6 Beyond that, DSMC can be employed to extract spatiotemporal autocorrelations along particle trajectories besides thermal fluctuations, both consistent with statistical thermodynamics. 7 Alternative particle Monte-Carlo schemes have been introduced in order to improve the efficiency of DSMC at low Knudsen numbers [8][9][10][11][12] or low Mach numbers. 13,14 One of the main issues that may hinder efficient particle based simulations arises due to large variations of the particle density throughout the physical domain.…”