Abstract
The mass flow rate of Poiseuille flow of rarefied gas through long ducts of two-dimensional cross-sections with arbitrary shape is critical in the pore-network modeling of gas transport in porous media. Here, for the first time, the high-order hybridizable discontinuous Galerkin (HDG) method is used to find the steady-state solution of the linearized Bhatnagar–Gross–Krook equation on two-dimensional triangular meshes. The velocity distribution function and its traces are approximated in piecewise polynomial spaces (of degree up to 4) on the triangular meshes and mesh skeletons, respectively. By employing a numerical flux that is derived from the first-order upwind scheme and imposing its continuity weakly on the mesh skeletons, global systems for unknown traces are obtained with fewer coupled degrees of freedom when compared to the original discontinuous Galerkin formulation. To achieve fast convergence to the steady-state solution, a diffusion-like equation for flow velocity, which is asymptotic-preserving into the fluid dynamic limit, is solved by the HDG simultaneously on the same meshes. The proposed HDG-synthetic iterative scheme is proved to be accurate and efficient. Specifically, for flows in the near-continuum regime, numerical simulations have shown that, to achieve the same level of accuracy, our scheme could be faster than the conventional iterative scheme by two orders of magnitude, also it is faster than the synthetic iterative scheme based on the finite difference discretization in the spatial space by one order of magnitude. In addition, the implicit HDG method is more efficient than an explicit discontinuous Galerkin gas kinetic solver, as well as the implicit discontinuous Galerkin scheme when the degree of approximating polynomial is larger than 2. The HDG-synthetic iterative scheme is ready to be extended to simulate rarefied gas mixtures and the Boltzmann collision operator.
Highlights
Accurate physical models and efficient numerical methods are needed to describe the gas flow spanning a wide range of rarefactions
To assess the accuracy and efficiency of the proposed scheme, our numerical results are compared with the discrete unified gas-kinetic scheme (UGKS) (DUGKS) solutions, which have been verified from the continuum to free-molecular flow regimes [22], or available data from literature
For the structured triangular mesh used, the Comparison between the hybridizable discontinuous Galerkin (HDG)-conventional iterative scheme (CIS), implicit DG (IDG)-CIS and Runge–Kutta DG (RKDG)-CIS for Poiseuille flow along a channel with square cross-section at δ = 10, where Itr is the number of iteration steps to satisfy the convergence criterion R < 10−5, Ndof denotes the number of degree of freedom involving in each method, t is the iterative time interval and tc is the CPU time
Summary
Accurate physical models and efficient numerical methods are needed to describe the gas flow spanning a wide range of rarefactions. Due to the rapid development of micro-electro-mechanical systems and the shale gas revolution in North America, extensive works have been devoted to constructing efficient deterministic schemes These methods often adopt a numerical quadrature to approximate the integration with respect to the molecular velocity on a discrete set of velocities [5]. In order to suppress the numerical diffusion errors, the size of spatial cell and time interval should be smaller than the mean free path and the mean collision time, respectively [11] For this reason, the deterministic technique becomes costly for near-continuum flows. Accurate and efficient numerical method for solving the gas kinetic equation is urgently needed to find the mass flow rate or apparent permeability of these pores, such that the permeability of the porous media can be obtained by the “Kirchhoff’s circuit law” based on the pore-network modeling.
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