A fast spectral method for the Boltzmann equation for monatomic gas mixtures
Introduction
There are a large number of applications for the Boltzmann equation (BE) in modeling rarefied gas flows frequently encountered in high-altitude aerodynamics, vacuum technologies, and microelectromechanical systems. While there is a great need for efficient and accurate methods for solving the BE, this is a challenge because the BE employs a molecular velocity distribution function (VDF) defined in a six-dimensional phase space to describe the system state, and the Boltzmann collision operator is highly complicated.
In 1912, Hilbert proposed obtaining approximate solutions to the BE by a series expansion in Knudsen number (Kn, defined as the ratio of the molecular mean free path to the characteristic flow length). The Hilbert expansion inspired Sone to develop an asymptotic analysis of the BE at small Kn, which helped reveal many interesting phenomena including the ‘ghost effect’ [1], [2]. In 1917, Chapman and Enskog (CE) obtained approximate solutions to the BE and calculated transport coefficients from first principles [3]. The first-order CE expansion recovers the Navier–Stokes equations, while the second- and third-order expansions lead to the Burnett and super-Burnett equations, respectively. Although the Burnett equations can be more accurate than the Navier–Stokes equations in some cases [4], they are rarely used nowadays due to their intrinsic instability to small-wavelength perturbations [5]. In 1949, by expanding the VDF into Gauss–Hermite polynomials, Grad proposed the moment method [6]. This has led to the development of regularized 13-moment [7] and 26-moment [8] equations, which have been successfully applied to rarefied gas flows up to in some circumstances. For large Kn, a great number of moments are necessary but the convergence rate with increasing moment number is rather slow [9]. Moreover, multiple new boundary conditions are needed, so this approach it is not suitable to simulate highly rarefied gas flows.
For moderate and highly rarefied gas flows it is necessary to directly solve the BE. The direct simulation Monte Carlo (DSMC) method is the prevailing numerical technique for this [10]. In DSMC, the VDF is represented by a number of simulated particles that move in the computational domain and collide according to stochastic rules. The simulated particles automatically concentrate in regions where the VDF has large values, and discontinuities/fine structures in VDFs can be captured without difficulty. This method is efficient for hypersonic flows, however, its stochastic nature makes it not well suited for unsteady and low-speed flows. For instance, for a gas flow with a Mach number 0.001 (typically in micro-devices), about 108 independent samples are needed to reduce the error to 1% when there are 100 simulated particles in a cell [11]. Also, DSMC is an inefficient method for near-continuum flows, since the spatial and temporal steps must be smaller than the molecular mean free path and collision time, respectively. Progress has been made to ease these difficulties: to resolve small macroscopic quantities, the information preserving [12] and low variance [13], [14] DSMC methods have been proposed; to simulate near-continuum flows, hybrid continuum-particle approaches [15], [16], [17], [18], [19], [20], [21], time relaxed Monte Carlo method [22], [23], and particle–particle hybrid methods [24], [25] have been proposed. Recently, a new multiscale method has also been developed to simulate flows in geometries with high aspect ratio [26], [27], [28].
Deterministic numerical methods [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], based on the direct discretization of the BE in phase space, have distinct advantages in resolving small signals and low-speed flows as they are free of noise. Also, implicit time-marching can be employed to enlarge the time step [29], [30], [41], [42], [43]. Furthermore, compared to continuum/particle coupling, deterministic methods can be coupled more efficiently to macroscopic equations [44] such as the Navier–Stokes/moment equations and equations from the asymptotic kinetic analysis [2]. Among these deterministic solvers, however, only a few have been applied to spatially-inhomogeneous problems. The numerical kernel method [29], [30], in which the VDF is expanded into Laguerre polynomials for velocity components parallel to solid surfaces and quadratic finite-element functions for normal velocity component, provides accurate numerical results for many one-dimensional problems. As it allows non-uniform discretization of the normal velocity component, VDF discontinuities at large Kn are captured: for linearized Poiseuille and thermal transpiration flows, accurate results are produced up to for hard-sphere molecules [45]. The conservative projection-interpolation method, in which uniform Cartesian velocity discretization is usually employed, and post-collision velocities are interpolated to closest neighbors, has been proposed to solve the BE with realistic intermolecular potentials in three-dimensional space [32], with a numerical error of the order (Δv is the velocity grid step, f is the VDF, and is the equilibrium VDF) [44]. Since VDFs at post-collision velocities are obtained by polynomial interpolation, this method is accurate at small Kn where the VDF is usually smooth (in a recent paper [46] an accuracy of has been demonstrated), but loses accuracy at large Kn where VDFs have discontinuities or steep variations. Discontinuous Galerkin methods, although able to capture VDF discontinuities, turn out to be feasible only for slightly rarefied subsonic gas flows [40] because of the high computational cost.
Recently, the fast spectral method (FSM) has been developed to numerically solve the BE for single-species monatomic gases [35], [47], [41], [42], [43]. It employs a Fourier–Galerkin discretization in velocity space, and handles binary collisions in the corresponding frequency space. The method is of spectral accuracy, and has a computational cost of , where N is the number of frequency nodes in each frequency direction and is the number of discrete solid angles. Since it allows non-uniform velocity discretizations, and the number of frequency nodes can be far smaller than the number of velocity nodes [42], [43], the FSM is good at dealing with highly rarefied gas flows, where large numbers of velocity nodes are used to capture discontinuities/fine structures in the VDF: for linearized Poiseuille and thermal transpiration flows, accurate numerical results have been obtained even for (note that even at such high Knudsen numbers, collision cannot be ignored). The FSM can be 50 times faster than the low variance DSMC method [42], and it is very efficient in dealing with linearized oscillatory flows [43] and Rayleigh–Brillouin scattering of light by rarefied gases [48], [49].
Deterministic numerical methods for the BE for mixtures of monatomic gases are even more scarce. Compared to the single-species BE, the non-unitary mass ratio between different molecular species poses an additional difficulty. Accurate numerical results have previously only been reported for hard-sphere molecules in some simple spatial configurations [50], [51], [52], [53], [54], [55]. For general cases, a multipoint conservative projection-interpolation method has been developed for an arbitrary ratio of molecular masses [56], but its accuracy is not known, especially at large Knudsen numbers. Recently, a spectral-Lagrangian method with a computational cost of has been proposed (where is the mass ratio of the heavier to the lighter species), and a normal shock wave in a binary mixture with a mass ratio of about 2 has been simulated [57]. As this method uses the same velocity discretization for each component, it cannot be applied to mixtures with large mass ratios, since the computational cost and storage are and times larger than that for the single-species BE, respectively. As an example, calculation of an argon–helium mixture with the mass ratio requires a 30-times increase in velocity grid points, and a 1000-times increase in computational time.
Because of the excellent performance of the FSM, in this paper we develop an efficient and accurate FSM for gas mixtures. The rest of this paper is organized as follows. In Section 2, the BE is introduced. The fast spectral approximation to the Boltzmann collision operator with general forms of the collision kernel is presented in Section 3, and its accuracy is assessed in Section 4 by comparing numerical results with analytical solutions for the spatially-homogeneous relaxation of pseudo-Maxwell molecules. In Section 5, an efficient algorithm is proposed to deal with gas mixtures of large molecular mass ratio. In Section 6, an iteration scheme is used to find the steady-state solutions of spatially-inhomogeneous problems, and various comparisons with results from the numerical kernel and DSMC methods are made for normal shocks and planar Fourier/Couette flows. A two-dimensional temperature-driven flow is also investigated. In Section 7, we conclude with a summary of our proposed numerical method, and outline future perspectives.
Section snippets
Boltzmann equation for a binary gas mixture
Consider a binary gas mixture of components A and B, where the mass of a molecule of component A is , and that of component B is . Without loss of generality, we assume . Let and be respectively the VDFs of the components A and B of the molecular velocity at spatial location and time t. In the absence of external forces, the BEs for a binary gas mixture of monatomic molecules take the following forms:
Fast spectral method for the cross-collision operators
The approximation of the self-collision operator (with special forms of the collision kernel) by using FSM has been studied extensively [34], [36], [41], [42]. Here we consider the numerical approximation of cross-collision operators with general forms of the collision kernel.
In FSM, VDFs are periodized on the velocity domain. For simplicity, we assume the velocity domains of and are the same, say, . The velocity bound L should be chosen as a compromise between the resolution
Accuracy analysis of the fast spectral method
We assess the accuracy of our newly-developed FSM by comparing its numerical results with the analytical Bobylev–Krook–Wu (BKW) solutions for the space-homogeneous relaxation of pseudo-Maxwell molecules when the collision kernel is . For simplicity, we take , and set the equilibrium temperature to be 2. The exact BKW solutions can be written as follows [59], [60]: where
An efficient algorithm for large mass ratios
In Section 4 we used the same velocity discretization for both VDFs. This is acceptable for mass ratios less than 2, but it becomes computationally inefficient for large mass ratios, as the computer memory usage of FSM is proportional to , while the computational cost is proportional to . Here we propose a more efficient algorithm to deal with large-mass-ratio problems.
Before starting, we demonstrate how the cross-collision operators are calculated in Section 4 when using the same
Numerical results for spatially-inhomogeneous problems
It is convenient to use dimensionless variables. Here, we normalize the spatial location x by the characteristic flow length ℓ, the molecular velocity v by the thermal speed of the A-component at the reference temperature , the density n by the reference molecular number density , and the VDF by . Then the BE (1) is recast to a dimensionless form, where the collision operator (2) with the collision kernel (6) is given by
Conclusions
We have developed a fast spectral method to solve the Boltzmann equation for binary mixtures of monatomic gases, with general forms of the collision kernel. The method employs Fourier–Galerkin discretization in velocity space, and handles binary collisions in the corresponding frequency space. The method has spectral accuracy, and can deal with mixtures with large molecular mass ratios at a computational cost of , which is much smaller than the conventional spectral method's cost
Acknowledgements
The authors thank the reviewers of this paper for their valuable suggestions. This work is financially supported by the UK's Engineering and Physical Sciences Research Council (EPSRC) under grants EP/I036117/1 and EP/I011927/1.
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