Octant flux splitting information preservation DSMC method for thermally driven flows

Abstract

We present the octant flux splitting DSMC method as an efficient method for simulating non-equilibrium flows of rarefied gas, particularly those arising from thermal loading. We discuss the current state-of-the-art flux splitting IP-DSMC technique and show that it fails to capture the shear stresses created by thermal gradients. We present the development of the octant flux splitting IP-DSMC as well as degenerate 2D, 1D, and 0D forms and apply the method to a number of problems including thermal transpiration, with satisfactory results.

Introduction

A number of interesting phenomena may occur in the presence of thermal loading in a rarefied gas. In 1825, Fresnel noticed that small bodies suspended in a gas would sometimes move when exposed to light—something later studied in detail by Crookes and others using a variety of the now familiar radiometers [1]. In 1879, Osborne Reynolds gave the name thermal transpiration to the effect he observed wherein gas was pumped through capillaries or porous plugs subject to a temperature gradient. In that same year, James Clerk Maxwell proposed a theoretical explanation for the mechanism of both thermal transpiration and the radiometric effect [2]. Recently, a number of micro- and nanoscale systems have sought to leverage thermally driven rarefied gas dynamic phenomena for pumping [3], [4], [5], [6], propulsion [7] and sensing [8]. Although the physical mechanisms are understood, detailed modeling of such systems is complicated by the difficulty involved in modeling low-speed non-equilibrium rarefied gas systems.

The degree of rarefaction of a gas is quantified by the Knudsen number which is the ratio of the mean free path between intermolecular collisions (λ) and some characteristic length, L, of the flow; i.e., Kn = λ/L. At small Kn, the intermolecular collisions dominate – resulting in a diffusive nature that can be accurately described by continuum models, e.g., the Euler and Navier–Stokes equations [9], [10]. At larger Kn, either due to decreasing densities (and correspondingly larger mean free paths) or smaller characteristic lengths, the contribution of intermolecular collisions will diminish until conservation equations cease to form a closed set at Kn > 0.1 and the continuum description breaks down [9]. Within the transition regime, 0.1 < Kn < 10, intermolecular collisions remain important – but not dominant – and the molecular nature of the gas must be considered. Finally, the free molecular regime, Kn > 10, is characterized by extremely rare intermolecular collisions and ballistic transport.

The fundamental equation describing the molecular nature of gases is the Boltzmann transport equation (BTE). One method of solving the BTE is a direct mathematical approach using distribution functions as the primary variables. However, the solution of the complete BTE has proven very difficult, requiring significant computational expense evaluating distributions and collision integrals. This has led to the development of various linearized models and simplified collision integrals (BGK, S-model, etc.) suitable for a range of slightly non-equilibrium problems [11]. An alternative approach is to model the behavior of the individual molecules directly, leading to a molecular dynamics (MD) approach. However, the large number of molecules (and the associated computational expense) required to adequately model a dilute gas precludes the MD approach. The direct simulation Monte Carlo largely overcomes this difficulty by using simulation molecules, each representing a large number of real molecules. This has the additional advantage that collisions—partners and post-collision behavior—may be treated in a probabilistic sense for further computational efficiency [9].

Although the DSMC has been used to model a number of thermally driven problems, including transpiration [6], such problems typically require a large number of samples. Low-speed flows, common in both small-scale devices and thermally driven systems, pose a particular challenge to efficient DSMC simulation as a large number of samples are required to control statistical noise. For example, air at standard temperature and pressure (STP) is composed of a myriad of molecules, each with velocities on the order of 500 m/s which are essentially independent of the stream velocity. A DSMC simulation of a 1 m/s flow of air at STP would require over 8 × 10⁶ independent samples to resolve the stream velocity within 1% [12].

The slow convergence of the DSMC simulations of low-speed flows has motivated the development of specialized DSMC techniques. Pan et al. proposed a DSMC method which split the molecular velocities into two parts: thermal and flow. By considering only the flow component in sampling they succeeded in reducing scatter for certain isothermal flows [13]. In a subsequent paper, Pan et al. [14] developed a block model suitable for non-isothermal flows wherein “big molecules” with modified masses and collision cross sections are used to replace the conventional simulation molecules. However, as pointed out in [30], the molecular block model doesn’t preserve the same flow conditions and thus is not accurate. Chun and Koch recently proposed a heavily modified DSMC which adds “ghost” molecules and variable weighting of particles during collisions and sampling [15].

The obvious similarity in all of these methods is to separate the information due to thermal energy from that associated with the stream velocity. The information preserving DSMC (IP-DSMC), originally proposed by Fan and Shen, is another such method that achieves variance reduction by storing and propagating certain collective quantities (information) with each simulation molecule [16]. If we consider each simulation (DSMC) molecule to be a representative sample from a large set of real molecules (i.e., with a position and velocity that an individual real molecule could have), then the preserved (IP) quantities may be interpreted as approximations of the collective, or macroscopic, information of the ensemble of real molecules represented by the simulation particle. Within the simulation, preserved quantities are propagated by particle motion (DSMC velocity) and sampled to obtain the macroscopic averages for the ensemble of particles within the computational cells. The resulting samples reflect the reduced noise in the preserved quantities.

The initial IP-DSMC model presented by Fan and Shen preserved only the velocity, which proved adequate for the isothermal channel flow they considered [16]. Subsequently, the IP-DSMC has been extended to treat more general flows by preserving additional quantities—density and temperature—and by various models for the transport and conservation of mass, momentum, and energy [17], [18], [19], [20], [21], [22]. In all cases, an update step accounts for the effects of mass, momentum and energy transport not captured by simulation particle movement. The early development of these IP-DSMC update techniques, as reported in the literature, has been somewhat ad hoc, beginning with the intuitive formulation for the acceleration of the gas due to pressure gradients [18], [23]. For non-isothermal flows, the transport of energy must be treated carefully and several methods have been proposed with varying degrees of success and applicability, including: modified collision cross-sections [18], the introduction of “additional energies” to molecules migrating between cells [20], and fluxal terms derived from kinetic theory [19], [21]. Recently, Sun and Boyd derived update equations that provide a formal connection between the preserved quantities and Maxwell’s equation of change and proposed two models to complete the update formulae, namely the local thermal equilibrium (LTE) and flux splitting (FS) methods [22]. To date, the flux splitting IP-DSMC technique appears to be the most accurate and generally applicable of the IP-DSMC techniques.

The performance and accuracy of the various IP-DSMC techniques has been demonstrated with satisfactory results through a number of benchmark problems including Poiseuille, Rayleigh, and thermal and velocity Couette flows, as well as shock structure problems in argon gas [20], [22]. In this paper, we are interested in applying IP-DSMC techniques to low-speed thermally driven flows, e.g., thermal transpiration. In doing so, we will show that the current (flux splitting) IP-DSMC methods cannot adequately model such flows. We will present the development of an “Octant” flux splitting IP-DSMC (or OSIP-DSMC) technique for efficient modeling of low-speed non-equilibrium rarefied gas flows, especially those arising from thermal loading.

The balance of this paper is organized as follows: Section 2 a brief review of the current state-of-the-art IP-DSMC methods; Section 3 a description of thermal transpiration and IP-DSMC modeling. The failure of the flux splitting method leads to the development of the Octant Splitting IP-DSMC presented in Section 4. Section 5 reports on the results of benchmark problems as well a few example problems, including thermal transpiration in a microchannel and thermal cavity problems. Finally, we present our conclusions in Section 6.