Discrete unified gas kinetic scheme for flows of binary gas mixture based on the McCormack model

The discrete unified gas kinetic scheme (DUGKS) was originally developed for single-species flows covering all the regimes, whereas the gas mixtures are more frequently encountered in engineering applications. Recently, the DUGKS has been extended to binary gas mixtures of Maxwell molecules on the basis of the Andries–Aoki–Perthame kinetic (AAP) model [P. Andries et al., “A consistent BGK-type model for gas mixtures,” J. Stat. Phys. 106, 993–1018 (2002)]. However, the AAP model cannot recover a correct Prandtl number. In this work, we extend the DUGKS to gas mixture flows based on the McCormack model [F. J. McCormack, “Construction of linearized kinetic models for gaseous mixtures and molecular gases,” Phys. Fluids 16, 2095–2105 (1973)], which can give all the transport coefficients correctly. The proposed method is validated by several standard tests, including the plane Couette flow, the Fourier flow, and the lid-driven cavity flow under different mass ratios and molar concentrations. Good agreement between results of the DUGKS and the other well-established numerical methods shows that the proposed DUGKS is effective and reliable for binary gas mixtures in all flow regimes. In addition, the DUGKS is about two orders of magnitude faster than the direct simulation Monte Carlo for low-speed flows in terms of the wall time and convergent iteration steps

Numerical study of nonequilibrium gas flow in a microchannel with a ratchet surface

The nonequilibrium gas flow in a two-dimensional microchannel with a ratchet surface and a moving wall is investigated numerically with a kinetic method [Guo, Phys. Rev. E 91, 033313 (2015)]PLEEE81539-375510.1103/PhysRevE.91.033313. The presence of periodic asymmetrical ratchet structures on the bottom wall of the channel and the temperature difference between the walls of the channel result in a thermally induced flow, and hence a tangential propelling force on the wall. Such thermally induced propelling mechanism can be utilized as a model heat engine. In this article, the relations between the propelling force and the top wall moving velocity are obtained by solving the Boltzmann equation with the Shakhov model deterministically in a wide range of Knudsen numbers. The flow fields at both the static wall state and the critical state at which the thermally induced force cancels the drag force due to the active motion of the top wall are analyzed. A counterintuitive relation between the flow direction and the shear force is observed in the highly rarefied condition. The output power and thermal efficiency of the system working as a model heat engine are analyzed based on the momentum and energy transfer between the walls. The effects of Knudsen number, temperature difference, and geometric configurations are investigated. Guidance for improving the mechanical performance is discussed

Application of discrete unified gas kinetic scheme to thermally induced nonequilibrium flows

In this article, we present the applications of the discrete unified gas kinetic scheme (DUGKS) for simulating thermal induced non-equilibrium flows. Four different types of thermally induced flows, including the thermal creep flow, thermal edge flow, radiometric flow and temperature discontinuity induced flow have been simulated in a wide range of Knudsen numbers. The numerical results have been compared with direct simulation Monte Carlo (DSMC) solutions and show that the Shakhov model based DUGKS can be faithfully used for such thermally induced nonequilibrium flows. In particular, due to the asymptotic preserving property of the DUGKS, the flow features in the near continuum flows can be captured efficiently. The extremely low-speed character of such flows is also in favor of the current deterministic model equation solver

Nonlinear oscillatory rarefied gas flow inside a rectangular cavity

The nonlinear oscillation of rarefied gas flow inside a two-dimensional rectangular cavity is investigated on the basis of the Shakhov kinetic equation. The gas dynamics, heat transfer, and damping force are studied numerically via the discrete unified gas-kinetic scheme for a wide range of parameters, including gas rarefaction, cavity aspect ratio, and oscillation frequency. Contrary to the linear oscillation where the velocity, temperature, and heat flux are symmetrical and oscillate with the same frequency as the oscillating lid, flow properties in nonlinear oscillatory cases turn out to be asymmetrical, and second-harmonic oscillation of the temperature field is observed. As a consequence, the amplitude of the shear stress near the top-right corner of the cavity could be several times larger than that at the left-top corner, while the temperature at the top-right corner could be significantly higher than the wall temperature nearly in the whole oscillation period. For the linear oscillation with the frequency over a critical value, and for the nonlinear oscillation, the heat transfer from the hot to cold region dominates inside the cavity, which is contrary to the anti-Fourier heat transfer in a low-speed rarefied lid-driven cavity flow. The damping force exerted on the oscillating lid is studied in detail, and the scaling laws are developed to describe the dependency of the resonance and anti-resonance frequencies (corresponding to the damping force at a local maximum and minimum, respectively) on the reciprocal aspect ratio from the near hydrodynamic to highly rarefied regimes. These findings could be useful in design of the micro-electro-mechanical devices operating in the nonlinear flow regime

dugksFoam : An open source OpenFOAM solver for the Boltzmann model equation

A deterministic Boltzmann model equation solver called dugksFoam has been developed in the framework of the open source CFD toolbox OpenFOAM. The solver adopts the discrete unified gas kinetic scheme (Guo et al., 2015) with the Shakhov collision model. It has been validated by simulating several test cases covering different flow regimes including the one dimensional shock tube problem, a two dimensional thermal induced flow and the three dimensional lid-driven cavity flow. The solver features a parallel computing ability based on the velocity space decomposition, which is different from the physical space decomposition based approach provided by the OpenFOAM framework. The two decomposition approaches have been compared in both two and three dimensional cases. The parallel performance improves significantly using the newly implemented approach. A speed up by two orders of magnitudes has been observed using 256 cores on a small cluster. Program summary Program Title: dugksFoam Program Files doi:http://dx.doi.org/10.17632/zwn7t9cf5w.1 Licensing provisions: The MIT License Programming language: C++ External routines/libraries: OpenFOAM (http://www.openfoam.org) Nature of problem: Solving the Boltzmann equation with Shakhov model explicitly. Solution method: Discrete unified gas kinetic scheme (DUGKS) Restrictions: Symmetric boundary condition can only be applied at walls parallel to axis directions

Fast convergence and asymptotic preserving of the general synthetic iterative scheme

Recently the general synthetic iteration scheme (GSIS) was proposed for the Boltzmann equation [W. Su et al., J. Comput. Phys., 407 (2020), 109245], where various numerical simulations have shown that (i) the steady-state solution can be found within dozens of iterations at any Knudsen number K, and (ii) the solution is accurate even when the spatial cell size in the bulk region is much larger than the molecular mean free path, i.e., the Navier-Stokes solutions are recovered at coarse grids. The first property indicates that the error decay rate between two consecutive iterations decreases to zero along with K, while the second one implies that the GSIS asymptotically preserves the Navier-Stokes limit when K approaches zero. This paper is first dedicated to the rigorous proof of both properties. Second, several numerically challenging cases (especially the two-dimensional thermal edge flow) are used to further demonstrate the accuracy and efficiency of GSIS

Thermally induced rarefied gas flow in a three-dimensional enclosure with square cross-section

Rarefied gas flow in a three-dimensional enclosure induced by nonuniform temperature distribution is numerically investigated. The enclosure has a square channel-like geometry with alternatively heated closed ends and lateral walls with a linear temperature distribution. A recently proposed implicit discrete velocity method with a memory reduction technique is used to numerically simulate the problem based on the nonlinear Shakhov kinetic equation. The Knudsen number dependencies of the vortices pattern, slip velocity at the planar walls and edges, and heat transfer are investigated. The influences of the temperature ratio imposed at the ends of the enclosure and the geometric aspect ratio are also evaluated. The overall flow pattern shows similarities with those observed in two-dimensional configurations in literature. However, features due to the three-dimensionality are observed with vortices that are not identified in previous studies on similar two-dimensional enclosures at high Knudsen and small aspect ratios

Performance evaluation of the general characteristics based off-lattice Boltzmann scheme and DUGKS for low speed continuum flows

The general characteristics based off-lattice Boltzmann scheme proposed by Bardow et al. [1] (hereafter Bardow's scheme) and the discrete unified gas kinetic scheme (DUGKS) [2] are two methods that successfully overcome the time step restriction by the collision time, which is commonly seen in many other kinetic schemes. In this work, we first perform a theoretical analysis of the two schemes in the finite volume framework by comparing their numerical flux evaluations. It is found that the effect of collision term is considered in the evaluation of the cell interface distribution function in both schemes, which explains why they can overcome the time step restriction and can give accurate results even as the time step is much larger than the collision time. The difference between the two schemes lies in the treatment of the integral of the collision term when evaluating the cell interface distribution function, in which Bardow's scheme uses the rectangular rule while DUGKS uses the trapezoidal rule. The performance of the two schemes, i.e., accuracy, stability, and efficiency are then compared by simulating several two dimensional flows, including the unsteady Taylor–Green vortex flow, the steady lid-driven cavity flow, and the laminar boundary layer problem. It is observed that, DUGKS can give more accurate results than Bardow's scheme with a same mesh size. Furthermore, the numerical stability of Bardow's scheme decreases as the Courant–Friedrichs–Lewy (CFL) number approaches to 1, while the stability of DUGKS is not affected by the CFL number apparently as long as CFL<1. It is also observed that DUGKS is twice as expensive as the Bardow's scheme with the same mesh size

Heat and mass transfer of oscillatory lid-driven cavity flow in the continuum, transition and free molecular flow regimes

Although effective cooling of micro-electro-mechanical systems (MEMS) with oscillatory components is essential for reliable device operation, the role of oscillation on heat transfer remains poorly understood. In this work, heat and mass transfer of the oscillatory gas flow inside a square cavity is computationally studied by solving the Boltzmann model equation, i.e. the Shakhov model. The oscillation frequency of the lid and rarefaction and nonlinearity of the flow field are systematically investigated. Our results show that, when the oscillation frequency of the lid increases, the usual cold-to-hot heat transfer pattern for highly rarefied flow changes to hot-to-cold, which contradicts the well-known anti-Fourier (i.e. cold-to-hot) heat transfer in a non-oscillatory lid-driven cavity. In addition, the thermal convection will be dramatically enhanced by lid oscillation, which may play a dominant role in the heat transfer. Meanwhile, the average Nusselt number varies non-monotonically with the oscillation frequency, with the maximum occurring at the anti-resonance frequency. Finally, the average Nusselt number on the lid at various oscillation frequencies is found to reduce when the gas becomes more rarefied. These findings may be useful for the thermal design of MEMS