LOCAL GRID REFINEMENT APPROACH FOR LATTICE BOLTZMANN METHOD: DISTRIBUTION FUNCTION CONVERSION BETWEEN COARSE AND FINE GRIDS
-
摘要: 格子Boltzmann方法作为一种高效的介观计算流体力学方法在过去20多年里得到快速发展, 其相对较高的计算效率和灵活性使其可以适用于各种复杂流动的模拟. 然而标准的格子Boltzmann方法只能使用均匀的直角网格, 这种网格排布方式并不利于复杂流动的计算. 为此, 基于格子Boltzmann方法的局部网格加密算法在文献中被提出. 该算法需要在局部加密的界面处将粗细网格间的分布函数转换后交换. 目前分布函数的转换方式大多是在没有源项的情况下推导的, 而且现存考虑源项时转换公式的推导也都是基于Chapman-Enskog展开; 其推导过程相对复杂, 且需要对分布函数的非平衡态部分做一阶Chapman-Enskog近似, 这有可能会限制局部网格加密算法在高阶格子Boltzmann方法中的应用. 文章在忽略时空离散误差的前提下, 以保证连续分布函数变量以及物理松弛系数一致为基础, 构建了一套规范且简洁的粗细网格间在考虑任意源项时, 分布函数转换关系的推导过程, 该方法不依赖于Chapman-Enskog展开以及Chapman-Enskog近似, 且该方法既可以适用于单松弛碰撞模型也可以适用于多松弛碰撞模型. 此外, 还从理论上证明了, 保证粗细网格间非平衡态部分的一阶 Chapman-Enskog 近似一致, 便可以保证整个非平衡态部分的一致, 这将有助于扩展局部网格加密算法中转换关系的应用范围. 最后, 通过对强迫泰勒−格林涡流动、平板泊肃叶流中对流−扩散问题和顶盖驱动方腔流动进行数值模拟, 良好的数值结果证实了转换关系对复杂源项的适应性以及局部网格加密技术在处理复杂流动问题方面的优势. 同时, 通过对一维剪切波问题的模拟, 发现由局部网格加密引起的数值黏性与加密区域的选取有很大的关系.
-
关键词:
- 格子Boltzmann方法 /
- 局部网格加密 /
- 数据转换
Abstract: Lattice Boltzmann method, as an efficient mesoscopic computational fluid dynamics method, has been developed rapidly in the past two decades. Its relatively high computational efficiency and flexibility make it suitable for the simulation of various complex flows. However, due to its own limitations, the standard lattice Boltzmann method typically utilize uniform rectangular grid, which is not suitable for the simulation of complex flows. Therefore, local grid refinement based on the lattice Boltzmann method has been considered by many researchers. For this purpose, the distribution functions between coarse and fine grids need to be converted at the interface of coarse and fine grids. At present, most conversion methods of distribution function are derived without the presence of the source term, and the previously limited derivation of conversion formulas considering the source term was based on the first-order Chapman-Enskog expansion, which is relatively complex and may limit the application of local grid refinement algorithm in higher-order lattice Boltzmann methods. In this paper, we provide a concise derivation to relate the distribution functions between coarse and fine grids considering an arbitrary source term, based on consistency requirements of the distribution function of the continuous Boltzmann equation between coarse and fine grids. The proposed method is independent of the Chapman-Enskog expansion and Chapman-Enskog approximation, and can be applied to both single relaxation time and multiple relaxation time collision models. In addition, this paper also proves theoretically that the consistency of the first-order Chapman-Enskog approximation of the non-equilibrium distribution between the coarse and fine girds can ensure the consistency of the entire non-equilibrium distribution, which expands the applicability of the previous conversion relationship. Finally, these theoretical results are validated by numerical simulations of a forced Taylor-Green vortex flow, convection-diffusion in a planar Poiseuille flow and lid-driven cavity flow. The good numerical results confirm the adaptability of the conversion relation in the presence of complex source terms and the advantages of local grid refinement technology in dealing with complex flow problems. At the same time, through the simulation of one-dimensional shear wave problem, it is found that the numerical viscosity caused by local grid refinement has a great relationship with the selection of refinement region.-
Key words:
- lattice Boltzmann method /
- grid refinement /
- data conversion
-
图 4 强迫泰勒−格林涡中在$ (x/L,y/L)=(0.4,0.25) $点处各宏观量随时间的变化. (a) ~ (c)分别为BGK模型和MRT模型下的相对压力 $ P-P_0 $、水平方向速度 $ u_x $和黏性应力 $ \sigma_{xx} $与理论结果的对比
Figure 4. The variation of each macroscopic quantity with time at the point $ (x/L,y/L)=(0.4,0.25) $ in the forced Taylor-Green vortex. (a) ~ (c) are comparisons between the relative pressure $ P-P_0 $, velocity in the $ x $ direction $ u_x $ and viscous stress $ \sigma_{xx} $ under BGK model and MRT model, respectively, and the theoretical results
图 5 强迫泰勒−格林涡中各宏观量的剖线图($tU_0/L=1,\;y/L=0.25$, 两条黑色虚线间的区域为加密区域). (a) ~ (c)分别为BGK模型和MRT模型下相对压力 $ P-P_0 $、水平方向速度 $ u_x $和黏性应力 $ \sigma_{xx} $与理论结果的对比
Figure 5. Profile diagram of each macroscopic quantity of forced Taylor-Green vortex ($ tU_0/L=1,\;y/L=0.25 $, the area between the two black dashed lines is refined). (a) ~ (c) are comparisons between the relative pressure $ P-P_0 $, velocity in the $ x $ direction $ u_x $ and viscous stress $ \sigma_{xx} $ under BGK model and MRT model, respectively, and the theoretical results
图 7 平板泊肃叶流中对流−扩散问题(黑色虚线为加密界面). (a) ~ (d)分别为BGK模型下水平方向速度 $ u_x $、标量 $ \phi $、通量 $ J_x $和$ J_y $与理论结果的对比
Figure 7. Diffusion problem in a planar Poiseuille flow (the black dashed lines are the refinement interface). (a) ~ (d) represent the velocity in x direction $ u_x $, scalar $ \phi $, flux $ J_y $ and $ J_x $ compared with theoretical results under the BGK model
图 9 $ Re=1000 $时顶盖驱动方腔流的压力分布云图对比, (a) ~ (d)分别为UCG, UFG, EQ和UCG-L对应的压力分布云图 ((d)中的黑色直线为加密界面, 界面以上为加密区域)
Figure 9. When $ Re=1000 $, the contours of pressure of the lid-driven cavity flow are compared. (a) ~ (d) are the contours of pressure corresponding to UCG, UFG, EQ and UCG-L, respectively (the black line in (d) is the grid refinement interface, and the area above the black line is the refinement area)
图 10 $ Re=1000 $时顶盖驱动方腔流中无量纲速度的剖线图, UCG, UFG, EQ和UCG-L与参考数据的对比. (a) 沿垂直直线穿过几何中心的$ u_x/U_w $, (b)为其局部放大, (c) 沿水平直线穿过几何中心的$ u_y/U_w $, (d)为其局部放大
Figure 10. Profile diagram of dimensionless velocity in a lid-driven cavity flow when $ Re=1000 $, comparison of UCG, UFG, EQ and UCG-L with reference data. (a) $ u_x/U_w $ in a vertical straight line through the center of the geometry, (b) local amplification of (a), (c) $ u_y/U_w $ in a horizontal straight line through the center of the geometry, (d) local amplification of (c)
图 12 一维剪切波问题在不同加密区域下均匀网格和非均匀网格对应的数值黏性和物理黏性之比的分布情况(黑色虚线为加密界面). (a) ~ (d)分别为不同加密区域下UCG-L和UCG对应的数值黏性和物理黏性之比随$ y/H $变化的剖线图 (续)
Figure 12. For one-dimensional shear wave problem, the distribution of the ratio of numerical viscosity and physical viscosity corresponding to uniform and non-uniform grids in different refinement regions (the black dashed line is the refinement interface). (a) ~ (d) are the cross-section plots of the ratio of numerical viscosity and physical viscosity corresponding to UCG-L and UCG with $ y/H $ in different refinement regions, respectively (continued)
表 1 不同分辨率下程序演化1000步时CPU消耗时间对比
Table 1. Comparison of CPU consumption time when the program evolves 1000 steps at different resolutions
UCG UCG-L UFG CPU time/s 1.656 5.803 6.782 -
[1] Kerimo J, Girimaji SS. Boltzmann-BGK approach to simulating weakly compressible 3D turbulence: comparison between lattice Boltzmann and gas kinetic methods. Journal of Turbulence, 2007, 8: N46 [2] Lin CD, Xu AG, Zhang GC, et al. Double-distribution-function discrete Boltzmann model for combustion. Combustion and Flame, 2016, 164: 137-151 doi: 10.1016/j.combustflame.2015.11.010 [3] Shan XW, Chen HD. Lattice Boltzmann model for simulating flows with multiple phases and components. Physical Review E, 1993, 47(3): 1815 doi: 10.1103/PhysRevE.47.1815 [4] Dellar PJ. Electromagnetic waves in lattice Boltzmann magnetohydrodynamics. Europhysics Letters, 2010, 90(5): 50002 doi: 10.1209/0295-5075/90/50002 [5] Aidun CK, Clausen JR. Lattice-Boltzmann method for complex flows. Annual Review of Fluid Mechanics, 2010, 42: 439-472 doi: 10.1146/annurev-fluid-121108-145519 [6] Lallemand P, Luo LS, Krafczyk M, et al. The lattice Boltzmann method for nearly incompressible flows. Journal of Computational Physics, 2021, 431: 109713 doi: 10.1016/j.jcp.2020.109713 [7] Marié S, Ricot D, Sagaut P. Comparison between lattice Boltzmann method and Navier-Stokes high order schemes for computational aeroacoustics. Journal of Computational Physics, 2009, 228(4): 1056-1070 doi: 10.1016/j.jcp.2008.10.021 [8] Hui X, Pierre S. Optimal low-dispersion low-dissipation lbm schemes for computational aeroacoustics. Journal of Computational Physics, 2011, 230(13): 5353-5382 doi: 10.1016/j.jcp.2011.03.040 [9] Kuznik F, Obrecht C, Rusaouen G, et al. LBM based flow simulation using gpu computing processor. Computers & Mathematics with Applications, 2010, 59(7): 2380-2392 [10] Blair S, Albing C, Grund A, et al. Accelerating an mpi lattice Boltzmann code using openacc//Proceedings of the Second Workshop on Accelerator Programming Using Directives, 2015: 1-9 [11] Chen T, Wang LP, Lai J, et al. Inverse design of mesoscopic models for compressible flow using the Chapman-Enskog analysis. Advances in Aerodynamics, 2021, 3(1): 1-25 doi: 10.1186/s42774-020-00055-6 [12] 陈涛. 可压缩湍流介观模型设计及应用. [博士论文]. 北京: 北京大学, 2021Chen Tao. Design and application of mesoscopic models for compressible turbulent flow. [PhD Thesis]. Beijing: Peking University (in Chinese) [13] Filippova O, Hänel D. Grid refinement for lattice-BGK models. Journal of Computational Physics, 1998, 147(1): 219-228 doi: 10.1006/jcph.1998.6089 [14] Lin CL, Lai YG. Lattice Boltzmann method on composite grids. Physical Review E, 2000, 62(2): 2219 doi: 10.1103/PhysRevE.62.2219 [15] Dupuis A, Chopard B. Theory and applications of an alternative lattice Boltzmann grid refinement algorithm. Physical Review E, 2003, 67(6): 066707 doi: 10.1103/PhysRevE.67.066707 [16] Touil H, Ricot D, Lévêque E. Direct and large-eddy simulation of turbulent flows on composite multi-resolution grids by the lattice Boltzmann method. Journal of Computational Physics, 2014, 256: 220-233 doi: 10.1016/j.jcp.2013.07.037 [17] Rohde M, Kandhai D, Derksen JJ, et al. A generic, mass conservative local grid refinement technique for lattice-Boltzmann schemes. International Journal for Numerical Methods in Fluids, 2006, 51(4): 439-468 doi: 10.1002/fld.1140 [18] Yu DZ, Mei RW, Shyy W. A multi-block lattice Boltzmann method for viscous fluid flows. International Journal for Numerical Methods in Fluids, 2002, 39(2): 99-120 doi: 10.1002/fld.280 [19] Astoul T, Wissocq G, Boussuge JF, et al. Analysis and reduction of spurious noise generated at grid refinement interfaces with the lattice Boltzmann method. Journal of Computational Physics, 2020, 418: 109645 doi: 10.1016/j.jcp.2020.109645 [20] Astoul T, Wissocq G, Boussuge JF, et al. Lattice Boltzmann method for computational aeroacoustics on non-uniform meshes: A direct grid coupling approach. Journal of Computational Physics, 2021, 447: 110667 doi: 10.1016/j.jcp.2021.110667 [21] Peng Y, Shu C, Chew YT, et al. Application of multi-block approach in the immersed boundary–lattice Boltzmann method for viscous fluid flows. Journal of Computational Physics, 2006, 218(2): 460-478 doi: 10.1016/j.jcp.2006.02.017 [22] Chapman S, Cowling TG. The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases. Cambridge: Cambridge University Press, 1990 [23] Huang RZ, Wu HY. Multiblock approach for the passive scalar thermal lattice Boltzmann method. Physical Review E, 2014, 89(4): 043303 doi: 10.1103/PhysRevE.89.043303 [24] d'Humières D. Multiple–relaxation–time lattice Boltzmann models in three dimensions. Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2002, 360(1792): 437-451 doi: 10.1098/rsta.2001.0955 [25] Liou TM, Wang CS. Three-dimensional multidomain lattice Boltzmann grid refinement for passive scalar transport. Physical Review E, 2018, 98(1): 013306 doi: 10.1103/PhysRevE.98.013306 [26] Bhatnagar PL, Gross EP, Krook M. A model for collision processes in gases. i. Small amplitude processes in charged and neutral one-component systems. Physical Review, 1954, 94(3): 511 doi: 10.1103/PhysRev.94.511 [27] 师羊羊. 非平衡流动的高阶格子玻尔兹曼方法. [博士论文]. 哈尔滨: 哈尔滨工业大学, 2021Shi Yangyang. High order lattice Boltzmann method for non-equilibrium flows. [PhD Thesis]. Harbin: Harbin Institute of Technology, 2021 (in Chinese) [28] Shan XW, Yuan XF, Chen HD. Kinetic theory representation of hydrodynamics: A way beyond the Navier–Stokes equation. Journal of Fluid Mechanics, 2006, 550: 413-441 doi: 10.1017/S0022112005008153 [29] Guo ZL, Xu K, Wang RJ. Discrete unified gas kinetic scheme for all knudsen number flows: Low-speed isothermal case. Physical Review E, 2013, 88(3): 033305 doi: 10.1103/PhysRevE.88.033305 [30] He XY, Shan XW, Doolen GD. Discrete Boltzmann equation model for nonideal gases. Physical Review E, 1998, 57(1): R13 doi: 10.1103/PhysRevE.57.R13 [31] 李旭晖, 单肖文, 段文洋. 格子玻尔兹曼正则化碰撞模型的理论进展. 空气动力学学报, 2022, 40(3): 1-20 (Li Xuhui, Shan Xiaowen, Duan Wenyang. The theory progress on the regularized lattice Boltzmann collision models. Acta Aerodynamica Sinica, 2022, 40(3): 1-20 (in Chinese)LI Xuhui, Shan Xiaowen, Duan Wenyang. The theory progress on the regularized lattice Boltzmann collision models[J]. Acta Aerodynamica Sinica, 2022, 40(3):1-20 (in Chinese). [32] Guo ZL, Zheng CG, Shi BC. Discrete lattice effects on the forcing term in the lattice Boltzmann method. Physical Review E, 2002, 65(4): 046308 doi: 10.1103/PhysRevE.65.046308 [33] Krüger T, Varnik F, Raabe D. Second-order convergence of the deviatoric stress tensor in the standard Bhatnagar-Gross-Krook lattice Boltzmann method. Physical Review E, 2010, 82(2): 025701 doi: 10.1103/PhysRevE.82.025701 [34] Lallemand P, Luo LS. Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, galilean invariance, and stability. Physical Review E, 2000, 61(6): 6546 doi: 10.1103/PhysRevE.61.6546 [35] Dong ZQ, Wang LP, Peng C, et al. A systematic study of hidden errors in the bounce-back scheme and their various effects in the lattice Boltzmann simulation of viscous flows. Physics of Fluids, 2022, 34(9): 093608 doi: 10.1063/5.0106954 [36] Chai ZH, Shi BC, et al. Multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-diffusion equations: Modeling, analysis, and elements. Physical Review E, 2020, 102(2): 023306 [37] Chen SY, Peng C, Teng YH, et al. Improving lattice Boltzmann simulation of moving particles in a viscous flow using local grid refinement. Computers & Fluids, 2016, 136: 228-246 [38] Shi YY, Wu L, Shan XW. Accuracy of high-order lattice Boltzmann method for non-equilibrium gas flow. Journal of Fluid Mechanics, 2021, 907: A25 [39] Gendre F , Ricot D , Fritz G, et al. Grid refinement for aeroacoustics in the lattice Boltzmann method: A directional splitting approach. Physical Review E, 2017, 96(2): 023311 doi: 10.1103/PhysRevE.96.023311 [40] Chen HD, Filippova O, Hoch J, et al. Grid refinement in lattice Boltzmann methods based on volumetric formulation. Physica A: Statistical Mechanics and its Applications, 2006, 362(1): 158-167 doi: 10.1016/j.physa.2005.09.036 [41] Min HD, Peng C, Guo ZL, et al. An inverse design analysis of mesoscopic implementation of non-uniform forcing in mrt lattice Boltzmann models. Computers & Mathematics with Applications, 2019, 78(4): 1095-1114 [42] Shi BC, Guo ZL. Lattice Boltzmann model for nonlinear convection-diffusion equations. Physical Review E, 2009, 79(1): 016701 doi: 10.1103/PhysRevE.79.016701 [43] Bird RB. Transport phenomena. Appl. Mech. Rev., 2002, 55(1): R1-R4 doi: 10.1115/1.1424298 [44] Flekkøy EG. Lattice bhatnagar-gross-krook models for miscible fluids. Physical Review E, 1993, 47(6): 4247 doi: 10.1103/PhysRevE.47.4247 [45] Guo ZL, Shi BC, Wang NC. Fully lagrangian and lattice Boltzmann methods for the advection-diffusion equation. Journal of Scientific Computing, 1999, 14: 291-300 doi: 10.1023/A:1023273603637 [46] Zhang MX, Zhao WF, Lin P. Lattice Boltzmann method for general convection-diffusion equations: MRT model and boundary schemes. Journal of Computational Physics, 2019, 389: 147-163 doi: 10.1016/j.jcp.2019.03.045 [47] Zheng HW, Shu Chang, Chew YT. A lattice Boltzmann model for multiphase flows with large density ratio. Journal of Computational Physics, 2006, 218(1): 353-371 doi: 10.1016/j.jcp.2006.02.015 [48] Chai ZH, Zhao TS. Lattice Boltzmann model for the convection-diffusion equation. Physical Review E, 2013, 87(6): 063309 doi: 10.1103/PhysRevE.87.063309 [49] Guo ZL, Zheng CG, Shi BC. An extrapolation method for boundary conditions in lattice Boltzmann method. Physics of Fluids, 2002, 14(6): 2007-2010 doi: 10.1063/1.1471914 [50] Zou QS, He XY. On pressure and velocity boundary conditions for the lattice Boltzmann BGK model. Physics of Fluids, 1997, 9(6): 1591-1598 doi: 10.1063/1.869307 [51] AbdelMigid TA, Saqr KM, Kotb MA , et al. Revisiting the lid-driven cavity flow problem: Review and new steady state benchmarking results using gpu accelerated code. Alexandria Engineering Journal, 2017, 56(1): 123-135 doi: 10.1016/j.aej.2016.09.013 [52] Ladd AJC. Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation. Journal of Fluid Mechanics, 1994, 271: 285-309 doi: 10.1017/S0022112094001771 [53] Nie XB, Shan XW, Chen H. Galilean invariance of lattice Boltzmann models. Europhysics Letters, 2008, 81(3): 34005 doi: 10.1209/0295-5075/81/34005 [54] Wan DD, Wang GC, Chen SY. Numerical investigation of lid-driven deep cavity with local grid refinement of MRT-LBM. Journal of Beijing Institute of Technology, 2019, 28(3): 536-548 [55] Guzik SM, Weisgraber TH, Colella P, et al. Interpolation methods and the accuracy of lattice-Boltzmann mesh refinement. Journal of Computational Physics, 2014, 259: 461-487 doi: 10.1016/j.jcp.2013.11.037 [56] Eitel-Amor G, Meinke M, Schröder W. A lattice-Boltzmann method with hierarchically refined meshes. Computers & Fluids, 2013, 75: 127-139 [57] Lagrava D, Malaspinas O, Latt J, et al. Advances in multi-domain lattice Boltzmann grid refinement. Journal of Computational Physics, 2012, 231(14): 4808-4822 doi: 10.1016/j.jcp.2012.03.015 [58] Grad H. Note on n-dimensional hermite polynomials. Communications on Pure and Applied Mathematics, 1949, 2(4): 325-330 doi: 10.1002/cpa.3160020402 -