A unified stochastic particle Bhatnagar-Gross-Krook method for multiscale gas flows

Highlights

• A unified stochastic particle ESBGK (USP-ESBGK) method is proposed for multiscale modeling of gas flows.

• The USP-ESBGK method has second-order accuracy in both time and space in the continuum regime.

• The USP-ESBGK method is an accurate and efficient tool to simulate multi-scale gas flows.

Abstract

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 size 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 with second-order accuracy. 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, Taylor-Green vortex, and flow through a slit, demonstrated that it is a promising tool to simulate multiscale gas flows ranging from rarefied to continuum regime.

Introduction

Multiscale modeling of gas flows is attracting more and more attentions as a large number of gas flows encountered in modern engineering problems are inherently multiscale, especially in aerospace engineering [1], [2] and micro-electro-mechanical system (MEMS) [3]. One example is high-speed gas flows around a reentry vehicle. Assuming the characteristic length of the reentry vehicle is 1 m, the global Knudsen number (Kn, the definition is the ratio of the molecular mean free path to the characteristic length) ranges from ${10}^{-6}$ to ${10}^{-1}$ at the altitudes of 20–100 km, and correspondingly the gas flow changes from continuum to transition regimes. Furthermore, if local structures such as the sharp leading edge or the microstructures on the vehicle surfaces are considered, the local Kn number spans a wider range, which will introduce a variety of thermochemical nonequilibrium phenomena and affect the flow fields around the reentry vehicle significantly. To accurately simulate such kinds of multiscale gas flows is very challenging. Although computational fluid dynamics (CFD) methods based on the Navier–Stokes (NS) equation have been successfully applied to the continuum regime, they encounter physical limitations for the simulation of gas flows far from equilibrium.

On the other hand, the direct simulation Monte Carlo (DSMC) method [4] on the molecular level is applicable to the simulation of nonequilibrium gas flows. Theoretically, DSMC is valid for the whole range of flow regime, as it can be regarded as a particle simulation method of solving Boltzmann equation. While it is popularly applied in the transition and near-continuum regime, the direct application of it to continuum regime is quite expensive due to the limitation of time step and cell sizes. Hence, a straightforward way to construct a multiscale method is coupling DSMC method and a CFD scheme, e.g., DSMC-CFD hybrid method, where the rarefied and continuum flow regimes are solved by the DSMC and CFD methods, respectively [5], [6], [7], [8]. However, DSMC-CFD hybrid approaches suffer from difficulties because of the amalgamation of two fundamentally different types of solvers [9]. It is very subtle to exchange information at the interface between DSMC and CFD regions.

One promising strategy for multiscale modeling is to develop a consistent solver for the whole flow regimes [10], [11]. Among others, one typical progress in this direction is the unified gas-kinetic scheme (UGKS) proposed by Xu and Huang [12] and discrete unified gas-kinetic scheme (DUGKS) proposed by Guo etc. [13], [14], which have been successfully applied to a variety of multiscale gas flows [15], [16], [17]. For both continuum and rarefied regimes, these two methods compute the gaseous distribution functions through discrete molecular velocities. However, the required number of discrete velocities increases significantly with the Mach number of the gas flow, in order to recover the distribution function accurately. In contrast, the stochastic particle method mimics the distribution function using simulation particles in a probabilistic way like DSMC method. Therefore, the required number of simulation particles is independent of the Mach and Knudsen numbers. Recently, some researchers have made efforts to develop a particle-particle hybrid method, such as BGK-DSMC [18], [19], [20], [21] and Fokker-Planck-DSMC [22], [23], [24], [25] methods, where the stochastic particle methods based on BGK or Fokker-Planck model are employed for the continuum regime, while DSMC method is used for the rarefied regime. It is known that BGK [26], [27], [28], [29], [30], [31], [32], [33] or Fokker-Planck [34], [35], [36], [37], [38] model simplifies the collision term in the Boltzmann equation, so their corresponding particle methods can achieve much higher efficiency than DSMC in the continuum regime. Compared to the UGKS and DUGKS methods, particle-particle hybrid methods are more efficient for the simulation of high speed gas flows, especially when complex physical and chemical effects are taken into account.

The stochastic particle method based on the BGK model was proposed by Macrossan [28] and Gallis and Torczynski [29] independently. Recently, the application of this method has been extended to complex gas flows [30], [31], [32], [33]. Note that in the current stochastic particle BGK method particle movements and collisions are decoupled in one calculating time step as same as that in the DSMC method. Consequently, their transport coefficients, such as viscosity and thermal conductivity, will deviate from physical values significantly if the time step size is larger than the molecular mean collision time. As analyzed by Chen and Xu [39], a successful multiscale gas kinetic scheme needs to inherently couple convection and collision effects when large temporal-spatial discretization is used.

In the present paper, a unified stochastic particle algorithm based on the BGK model is proposed by coupling molecular convection and collisions. Our aim is to improve the accuracy of the current stochastic particle BGK method for large temporal-spatial discretization and to develop a unified multiscale particle method in the end. The remainder of this paper is organized as follows. In section 2, we first review the ellipsoidal statistical Bhatnagar–Gross–Krook (ESBGK) model and the related stochastic particle method. In section 3, we present the principle and algorithm of the proposed unified stochastic particle method for multiscale gas flows. At last, several applications of the proposed method for a wide range of Knudsen numbers and time step sizes are presented in section 4.