A hybrid approach to couple the discrete velocity method and Method of Moments for rarefied gas flows
Introduction
Non-equilibrium gas flows exist in many industrial applications and scientific research facilities, including micro-electro-mechanical systems (MEMS), mass spectrometry, low-pressure environments, vacuum pumps, high-altitude vehicles, and ultra-tight porous media. It represents a fundamental modelling challenge due to the non-equilibrium effects that may be counter-intuitive and cannot be described by the Navier-Stokes-Fourier (NSF) equations. The extent of the non-equilibrium state is usually measured by the Knudsen number, Kn, which is the ratio of the gas molecular mean free path to the characteristic macroscopic length scale of the flow. It is well known that the NSF equations work well in the continuum regime () but as the Knudsen number increases, non-equilibrium effects begin to dominate and the NSF equations gradually lose their accuracy and validity. It is believed that the NSF equations could produce certain main features of the flow for simple problems in the slip regime () [1], provided appropriate velocity-slip and temperature-jump wall boundary conditions are applied. However, care must be taken when thermal effects are present [2], [3], for example, the ghost effect will render the Fourier heat conduction law inaccurate even when the Knudsen number approaches zero [4]. When the Knudsen number becomes appreciable, e.g. in the transition regime () or the free molecular regime (), the conventional constitutive relationships for the NSF equations do not hold any more [5] and it is necessary to adopt kinetic descriptions.
The Boltzmann equation, which uses a one-particle velocity distribution function (VDF) to describe the gaseous state, is the fundamental equation for kinetic theory in all flow regimes under the assumptions of binary collisions and molecular chaos. It is usually solved by probabilistic methods such as the direct simulation Monte Carlo (DSMC) method [6], [7]. In DSMC, each simulated particle represents a large number of real molecules, so that a great number of computational operations can be reduced. Despite its stochastic nature [8], DSMC is efficient for solving high-speed rarefied gas flows with moderate to high Knudsen numbers. However, when the Mach or Knudsen number is small, the computational costs increase dramatically, especially in the slip- and early transition-regimes. Variants of DSMC, such as the low variance DSMC method [9], employ variance reduction techniques to reduce the statistical noise. However, they are more suitable for extremely low-speed flows and flows with very small thermal gradients. In contrast to the probabilistic approach, deterministic methods have been developed to solve the Boltzmann equation numerically. Well-known examples are the discrete velocity methods (DVMs), firstly developed by Goldstein, Sturtevant and Broadwell in 1989 [10]. In DVMs, the continuous molecular velocity space is approximated by a finite set of discrete velocity points. Using numerical quadrature rules, the macroscopic quantities can be calculated from velocity moments of the VDF.
In the past two decades, much effort has been devoted to constructing efficient and accurate deterministic schemes. Many full Boltzmann solvers have been developed based on the Fourier transform techniques for Maxwell molecules, such as Fourier spectral methods [11], [12] and fast spectral methods [13], [14], [15], [16], [17]. However, the complicated structure of the nonlinear collision operator makes them formidable for many practical applications. Hence, simplified kinetic model equations, such as the Bhatnagar-Gross-Krook (BGK) [18], ES-BGK [19], Shakhov [20] models, and various corresponding numerical solvers for these Boltzmann model equations, have been proposed to reduce the computational cost. The gas kinetic unified algorithm was developed and applied to study hypersonic flows for spacecraft re-entry problems [21], [22]. Asymptotic preserving schemes, especially the unified gas kinetic scheme and the discrete unified gas kinetic scheme, were also proposed for problems involving different flow regimes [23], [24]. The performance of the discrete unified gas kinetic scheme and traditional Godunov DVM, in terms of accuracy and efficiency, has been evaluated with a wide range of Knudsen numbers [25]. Recently, an accurate and efficient discontinuous Galerkin method has been developed, which is faster than the conventional iterative scheme by two orders of magnitude [26].
Even though much research has been devoted to reducing the computational cost of the Boltzmann solvers, the multidimensional nature of the VDF and complicated structure of the nonlinear collision operator still poses a real challenge for them to be used for practical applications. Therefore, many macroscopic equations have been derived to describe the rarefied gas dynamics beyond the NSF level, such as Burnett [27], super-Burnett equations [28], Grad 13 [29], [30] and regularized 13/26 moment (R13/R26) equations [31], [32]. Among these macroscopic methods, the moment method (MM), proposed by Grad in 1949 [29], [30], has been demonstrated to be a potential engineering design tool for non-equilibrium flows in the transition regime [33], [34], [35], [36], [37].
In the MM, Grad [30] derived a set of governing equations for the shear stress and heat flux from the Boltzmann equation and closed this set of 13 moment equations (G13) by expanding the VDF into Hermite polynomials [29]. Struchtrup and Torrilhon [38], [39], [40] obtained the R13 equations by regularizing this set of equations with the aid of a Chapman-Enskog (CE) expansion [27]. Gu and Emerson [41] and Torrilhon and Struchtrup [42] obtained a set of wall boundary conditions (WBCs) for the R13 equations using Maxwell's kinetic WBCs [43]. The R13 equations improved the performance of the G13 moment equation for a Knudsen number up to 0.25. However, they cannot provide a sufficiently accurate description of the Knudsen layer [44], [45]. To improve the accuracy of MM further, Gu and Emerson [32] extended the theory and derived the regularized 26 (R26) moment equations, which have found a wide range of applications in the study of channel flow [32], [33], thermal transpiration flow [34] and gas flows in porous media [35]. Recently, the R26 moment equations have been applied to investigate rarefied effects on low Reynolds number flow past a circular cylinder in the slip and transition regime. An existence criterion has been discovered for the vortices expressed in a Reynolds number vs Knudsen number diagram [36].
The method of moments, which was originally proposed by Grad [30] as an approximate solution procedure to the Boltzmann equation, bridges the gap between hydro-thermodynamics and kinetic theory. In this approach, the Boltzmann equation is satisfied in a certain average sense rather than at the velocity distribution function level. However, the moment equations lose their accuracy close to the wall where gas molecules collide with the wall more frequently than among themselves and near-wall non-equilibrium effects are strong. Furthermore, the wall boundary conditions based on the truncated VDF are no longer adequate as the Knudsen number increases.
To extend the application of the macroscopic equations, various hybrid macroscopic and microscopic approaches have been proposed. Normally, the macro- and microscopic coupling approaches are developed based on the hybrid NSF/DSMC algorithms [46], [47]. They have been used in many applications and proved to be more efficient than the full DSMC method. Espinoza et al. [48] first developed an open-source CFD-DSMC hybrid code for high speed flows within the OpenFOAM framework. Li et al. [49] compared the NSF/DSMC hybrid results with full DSMC results for transitional hypersonic flow around a cylinder. Deschenes and Boyd [50] have recently devised a hybrid approach to investigate non-equilibrium effects by determining the breakdown location in hypersonic flows. Wijesinghe and Hadjiconstantinou [51] summarised the published work on coupling schemes at the interface and pointed out the overall considerations involved, such as the statistical noise. The data transfer from NSF systems to DSMC can be achieved by introducing a Chapman-Enskog velocity distribution.
The foregoing hybrid algorithms are mainly based on the NSF equations and DSMC method. However, the VDFs reconstructed from the NSF results are not accurate enough and the statistical noise inherent in DSMC also affects the numerical precision. The moment method can be used to extend the validity of continuum-hydrodynamic models and the more moments used in the Hermite polynomials, the more accurately the VDF can be reconstructed. Furthermore, there is no statistical noise involved, since both DVM and the MM are deterministic methods. Hence, we employ the MM and DVM to investigate rarefied gas flows at both the macroscopic and microscopic levels. It is straightforward to couple the DVM with different order of moment equation systems, which yields hybrid DVM/NSF, DVM/R13 or DVM/R26 algorithms.
In the present study, a hybrid discrete velocity method/moment method (DVM/MM) approach is proposed and the performance of the hybrid method, in terms of accuracy and efficiency, are evaluated. The remaining part of this paper is organised as follows. We first make a brief introduction of the Boltzmann equation in Section 2, which can be used to produce benchmark solutions, followed by an overview of the moment method in Section 3. The hybrid DVM/MM numerical algorithm will be outlined in Section 4. Based on the hybrid method, detailed results for pressure-driven Poiseuille flow, lid-driven cavity flow, thermally induced flow, and flow past a square cylinder will be given in Section 5, followed by discussion and conclusions in Section 6.
Section snippets
The kinetic equation and the moments
Kinetic theory accounts for a molecule's movement and interaction through a molecular velocity distribution function, , where x and C are the position and velocity vectors, respectively, of a molecule at time, t, and gives the number of molecules whose velocities lie within in a volume element, . For convenience, a mass distribution function is used in the present study and is defined by where m is the mass of a molecule. The Boltzmann equation,
An overview of the moment method
As the traditional hydrodynamic quantities of density, ρ, bulk velocity, , and temperature, T, correspond to the first five lowest-order moments of the VDF, their governing equations can be readily derived from the Boltzmann equation. They represent mass, momentum, and energy conservation laws, respectively [31]: The pressure p is given by the ideal gas law . However, the stress tensor, ,
Hybrid DVM/MM algorithm
When the gas is in a rarefied state or the size of a device is comparable to the mean free path, there is not a sufficient number of collisions among the gas molecules to reach the equilibrium state. Rather, the gas molecules collide with the wall more frequently than among themselves. The kinetic equation described in Section 2 is required to accurately capture the detailed non-equilibrium effects. When the gas is far away from the wall, the wall effect on the gas becomes weak and the gas will
Numerical test cases and discussion
In this section, the proposed hybrid algorithm is employed to study pressure-driven Poiseuille flow, a lid-driven cavity flow, 2D flow induced by a hot micro-beam in a rectangular chamber, and flow past a square cylinder. In all tested cases, the DVM and the hybrid DVM/MM approaches share the same spatial meshes and the gas medium is modelled as an argon gas. The viscosity is obtained from Sutherland's law [63]: where the reference viscosity and temperature are,
Conclusions
In the present work, a hybrid algorithm coupling the discrete velocity method (DVM) and the moment method (MM) has been developed and validated for rarefied gas flows in the early transition regime. Four different test cases involving pressure-driven Poiseuille flow, lid-driven cavity flow, thermal edge flow, and the flow past a square cylinder, have been presented to show the accuracy and capability of the proposed method.
Limitations were observed in the moment method. The results show that,
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Authors declare that they have no conflict of interest.
Acknowledgements
This work was supported by the United Kingdom Engineering and Physical Sciences Research Council (EPSRC) under grants EP/N016602/1, EP/R029326/1 and EP/R029369/1, Science and Technology Facilities Council (STFC) under grant ST/R006733/1 and the Computational Science Centre for Research Communities (CoSeC).
The financial support to W. Yang from University of Strathclyde and the Chinese Scholarship Council (CSC) during his visit to the UK are greatly acknowledged. W. Yang would also like to thank
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