Fig. 1. A quarter of the 3D three-level Sierpinski carpet (halved in the y and z directions, and a half of the central xz cross-section is visible here) is modeled due to the symmetric property, the methane flow is driven in the x direction by a small pressure drop p_in − p_out = 0.01p_out, the porosity is (27^3 − 2107)/27^3 , the dominant pore size h is equal to the size of the smallest elementary black block, the total sizes are 27h × 13.5h × 13.5h, the xy plane (marked by the blue line at Z = z/h ≡ 12) crossing the top elementary blocks at their middle is used for the convergence study of flow velocity distribution.
Fig. 2. Comparison between the DVM and DSBGK results of the normalized flow velocity u(X) along the line of Z ≡ 12 and Y ≡ 4.5 obtained by using different spatial grids, and using different-order upwind schemes (only for the DVM simulations), Kn = 1. Due to the almost symmetric property of u(X) with respect to X = 0, for clarity, we only show the DVM and DSBGK results for X < 0 and X > 0, respectively.
The notation L1h4V10 denotes the DSBGK simulation, where the thickness of each extra fluid layer added to the inlet and outlet is L = 1h, the dominant pore size h is resolved by 4 uniform grids/cells (i.e. 4^3 spatial grids for each elementary block) and the molecular velocity grids are 10 simulated molecules per cell; additionally, the notation is slightly changed to L1h4V8^3S2 in the DVM simulation if it uses 8^3 velocity grids at each spatial grid and the second-order upwind scheme to approximate the spatial derivative.
Fig. 3. Comparison between the DVM and DSBGK results along the line of Z ≡ 12 and Y ≡ 4.5 obtained by using different molecular velocity grids, Kn = 1. Note that the noticeable difference between the DVM and DSBGK results of fine velocity grids is due to using h8 in the DVM, which is insufficient as shown in Fig. 2.
Fig. 4. Comparison between the converged DSBGK results (black lines) obtained by using L1h8V10 and the best DVM results (blue lines) that we can afford by using L1h32V16^3S1 along the line of Z ≡ 12 and Y ≡ 4.5 for u(X) and the line of Z ≡ 12 and X ≡ −1.5 for u(Y ), Kn = 1.
The current DVM simulation is run by using 64 compute nodes (each has 128 GB of memory for 2 Message Passing Interface (MPI) processes and 12 Open Multi-Processing (OpenMP) threads per MPI process) on the UK national supercomputer ARCHER and has reached the limit of our compute capability. In sharp contrast, the DSBGK simulation can run on an ordinary laptop using about 2.624 GB and 2 CPU cores.
Fig. 5. Permeability κ(Kn) computed by the DVM and DSBGK methods. The intrinsic permeability at Kn = 0 computed by the LBM with L0h20 is about 0.06159 mD. The elementary block size is h = 10 nm here.
There are several simulation methods for the rarefied gas flows. The direct simulation Monte Carlo (DSMC) method independently proposed by Graeme A. Bird in 1960 is the well-recognized method of equivalently solving the Boltzmann equation but its computational cost at low speed (or low Mach number in general) is unfortunately prohibitive due to low signal-to-noise ratio (e.g., the flow speed is about 0.001 m/s while the random gas molecular speed or sound speed at standard temperature is larger than 300 m/s). The discrete velocity method (DVM) first proposed by James E. Broadwell in 1964 is a deterministic method and thus is noise-free. But, at each spatial grid, it needs a large number of velocity grids to discretize the unbounded molecular velocity space, which makes its three-dimensional simulations very costly and most of its publications limited to two-dimensional demonstrations. Additionally, there is another DSBGK method (i.e., direct simulation BGK method) independently proposed by Jun Li in 2009.
Among various applications, the rarefied gas flow through nanopores is very challenging to simulate due to high Knudsen number, low speeds and complicated pore geometries. Here, a comprehensive study of the grid convergence that determines the simulation accuracy is conducted to show the numerical performance. To get results of the same accuracy by using fine spatial and molecular velocity grids, the DSBGK method is orders of magnitude cheaper than the traditional DVM in terms of the CPU time and memory usage. While the DVM requires a high-performance computing (HPC) facility to meet the memory demand of about 767 gigabytes for simulating this representative 3D flow problem, the DSBGK simulation can run on an ordinary laptop using about 2.624 gigabytes. The DSBGK method therefore has unique advantages in studying low-speed, rarefied gas flows (e.g. Kn > 0.1) through complicated 3D porous geometries. Note that the DVM adopted in this study is more efficient than the discrete unified gas kinetic scheme (DUGKS) that is developed from the UGKS. Additionally, in comparison with the low variance deviational simulation Monte Carlo (LVDSMC) method, the DSBGK method can save computational cost by using a few orders of magnitude less samples to reduce the statistical noise. Thus, the DSBGK method is orders of magnitude cheaper than not only the DVM but all other possible methods in simulating real low-speed rarefied gas flows.