A high-order off-lattice Boltzmann method for incompressible flows

MA Chao; WU Jie

doi:10.7638/kqdlxxb-2021.0243

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A high-order off-lattice Boltzmann method for incompressible flows

更新时间：2026-05-28

- A high-order off-lattice Boltzmann method for incompressible flows
- Vol. 40, Issue 3, Pages: 65-74(2022)
- 作者机构：
  
  南京航空航天大学航空学院,南京,210016
- 作者简介：
- 基金信息：
- DOI：10.7638/kqdlxxb-2021.0243
  CLC： O357.1
- Published：2022
- 稿件说明：
移动端阅览
马超, 吴杰. 不可压缩流动的高精度非标准格子玻尔兹曼方法[J]. 空气动力学学报, 2022, 40(3): 65−74.

MA Chao, WU Jie. A high-order off-lattice Boltzmann method for incompressible flows[J]. 2022, 40(3): 65-74.
马超, 吴杰. 不可压缩流动的高精度非标准格子玻尔兹曼方法[J]. 空气动力学学报, 2022, 40(3): 65−74. DOI： 10.7638/kqdlxxb-2021.0243

MA Chao, WU Jie. A high-order off-lattice Boltzmann method for incompressible flows[J]. 2022, 40(3): 65-74. DOI： 10.7638/kqdlxxb-2021.0243.

摘要

首先回顾了高精度非标准格子玻尔兹曼方法的发展历程，基于高精度通量重构格式，发展了一种通量重构格子玻尔兹曼方法（FRLBM）。采用两种求解方法：一种是将碰撞项隐式处理，直接求解离散速度玻尔兹曼方程（直接法）；另一种是先执行碰撞步，再求解纯对流方程（分步法）。通过收敛性研究，比较了这两种方法的精度和稳定性。研究结果表明，在小时间步长下两种方法误差近似，都能取得高阶精度；然而当时间步长增大，直接法误差几乎不变，分步法误差出现明显上升。由此表明，当取得近似误差时，直接法可以采用较大时间步长，计算效率更高，且直接法的稳定性略占优。接着通过模拟顶盖驱动方腔流验证了FRLBM捕捉流场细节的能力，并且比较了基于半隐格式的显式方法和一阶、二阶及三阶隐式-显式Runge-Kutta格式的时间离散，在不同雷诺数下的最大允许Courant-Friedrichs-Lewy数，数值结果表明二阶隐式-显式Runge-Kutta格式效果最优。最后数值模拟了圆柱绕流，验证了FRLBM计算复杂外形绕流的可靠性。

Abstract

This paper firstly reviews the development of the high-order off-lattice Boltzmann method (OLBM). Based on the high-order flux reconstruction scheme (FR)

we develop a high-order flux reconstruction lattice Boltzmann method (FRLBM). Two numerical methods are adapted. One is to directly solve the discrete velocity Boltzmann equation with collision term treated implicitly (direct method) and the other is to sequencially implement collision step and the convection term (two-step method). Through the convergence study

the accuracy and stability of the two methods are compared. It is found that

when the time step is small

the errors of the two methods are in similar order

and the high-order accuracy can be obtained. However

when the time step is increased

the error of the direct method is almost unchanged

while the error of the two-step method increases significantly. The results indicate that the direct method can use a larger time step to obtain higher accuracy to abtain the reasonable accuracy in comparing with two step method. Meanwhile

the direct method also has better stability. Then

the ability of FRLBM to capture the details of the flow field is verified by simulating the lid-driven cavity flow. In addition

we compare the maximum Courant-Friedrichs-Lewy (CFL) numbers at different Reynolds numbers by semi-implicit time-marching scheme

first

second and third-order implicit-explicit (IMEX) Runge-Kutta schemes

respectively. The numerical results indicate the second-order IMEX performs best. Finally

the numerical simulation of the flow over a circular cylinder verifies the reliability of the FRLBM for calculation of the flow around the body with complex geometry.

关键词

Keywords

references

HE X Y, LUO L S. Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation[J]. Physical Review E, 1997, 56: 6811-6817. DOI: 10.1103/PhysRevE.56.6811

MEI R W, SHYY W. On the finite difference-based lattice Boltzmann method in curvilinear coordinates[J]. Journal of Computational Physics, 1998, 143(2): 426-448. DOI:10.1006/jcph.1998.5984

CHEN H D. Volumetric formulation of the lattice Boltzmann method for fluid dynamics: Basic concept[J]. Physical Review E, 1998, 58(3): 3955-3963. DOI:10.1103/physreve.58.3955

LEE T, LIN C L. A characteristic Galerkin method for discrete Boltzmann equation[J]. Journal of Computational Physics, 2001, 171(1): 336-356. DOI:10.1006/jcph.2001.6791

HE X Y, LUO L S, DEMBO M. Some progress in lattice Boltzmann method. Part I. Nonuniform mesh grids[J]. Journal of Computational Physics, 1996, 129(2): 357-363. DOI:10.1006/jcph.1996.0255

SHU C, NIU X D, CHEW Y T. Taylor-series expansion and least-squares-based lattice Boltzmann method: Two-dimensional formulation and its applications[J]. Physical Review E, Statistical, Nonlinear, and Soft Matter Physics, 2002, 65(3 Pt 2B): 036708. DOI:10.1103/PhysRevE.65.036708

LIN X J, WU J, ZHANG T W. A mesh-free radial basis function-based semi-Lagrangian lattice Boltzmann method for incompressible flows[J]. International Journal for Numerical Methods in Fluids, 2019, 91(4): 198-211. DOI:10.1002/fld.4749

SHI X, LIN J Z, YU Z S. Discontinuous Galerkin spectral element lattice Boltzmann method on triangular element[J]. International Journal for Numerical Methods in Fluids, 2003, 42(11): 1249-1261. DOI:10.1002/fld.594

DÜSTER A, DEMKOWICZ L, RANK E. High-order finite elements applied to the discrete Boltzmann equation[J]. International Journal for Numerical Methods in Engineering, 2006, 67(8): 1094-1121. DOI:10.1002/nme.1657

HEJRANFAR K, EZZATNESHAN E. A high-order compact finite-difference lattice Boltzmann method for simulation of steady and unsteady incompressible flows[J]. International Journal for Numerical Methods in Fluids, 2014, 75(10): 713-746. DOI:10.1002/fld.3916

HEJRANFAR K, EZZATNESHAN E. Implementation of a high-order compact finite-difference lattice Boltzmann method in generalized curvilinear coordinates[J]. Journal of Computational Physics, 2014, 267: 28-49. DOI:10.1016/j.jcp.2014.02.030

HEJRANFAR K, SAADAT M H, TAHERI S. High-order weighted essentially nonoscillatory finite-difference formulation of the lattice Boltzmann method in generalized curvilinear coordinates[J]. Physical Review E, 2017, 95(2-1): 023314. DOI:10.1103/physreve.95.023314

SUN Y X, TIAN Z F. High-order upwind compact finite-difference lattice Boltzmann method for viscous incompressible flows[J]. Computers Mathematics With Applications, 2020, 80(7): 1858-1872. DOI:10.1016/j.camwa.2020.08.014

LI W D. High order spectral difference lattice Boltzmann method for incompressible hydrodynamics[J]. Journal of Computational Physics, 2017, 345: 618-636. DOI:10.1016/j.jcp.2017.05.039

GUO Z L, ZHAO T S. Explicit finite-difference lattice Boltzmann method for curvilinear coordinates[J]. Physical Review E, 2003, 67(6): 066709. DOI:10.1103/physreve.67.066709

PIERACCINI S, PUPPO G. Implicit-explicit schemes for BGK kinetic equations[J]. Journal of Scientific Computing, 2007, 32(1): 1-28. DOI:10.1007/s10915-006-9116-6

MIN M S, LEE T. A spectral-element discontinuous Galerkin lattice Boltzmann method for nearly incompressible flows[J]. Journal of Computational Physics, 2011, 230(1): 245-259. DOI:10.1016/j.jcp.2010.09.024

ZADEHGOL A, ASHRAFIZAADEH M, MUSAVI S H. A nodal discontinuous Galerkin lattice Boltzmann method for fluid flow problems[J]. Computers Fluids, 2014, 105: 58-65. DOI:10.1016/j.compfluid.2014.09.015

KRÄMER A, KÜLLMER K, REITH D, et al. Semi-Lagrangian off-lattice Boltzmann method for weakly compressible flows[J]. Physical Review E, 2017, 95(2-1): 023305. DOI:10.1103/PhysRevE.95.023305

KRÄMER A, WILDE D, KÜLLMER K, et al. Lattice Boltzmann simulations on irregular grids: Introduction of the NATriuM library[J]. Computers Mathematics With Applications, 2020, 79(1): 34-54. DOI:10.1016/j.camwa.2018.10.041

VINCENT P E, CASTONGUAY P, JAMESON A. A new class of high-order energy stable flux reconstruction schemes[J]. Journal of Scientific Computing, 2011, 47(1): 50-72. DOI:10.1007/s10915-010-9420-z

YU M L, WANG Z J, LIU Y. On the accuracy and efficiency of discontinuous Galerkin, spectral difference and correction procedure via reconstruction methods[J]. Journal of Computational Physics, 2014, 259: 70-95. DOI:10.1016/j.jcp.2013.11.023

HE X Y, LUO L S. Lattice Boltzmann model for the incompressible navier-stokes equation[J]. Journal of Statistical Physics, 1997, 88(3-4): 927-944. DOI:10.1023/B: JOSS.0000015179.12689.e4

ROE P L. Approximate Riemann solvers, parameter vectors, and difference schemes[J]. Journal of Computational Physics, 1981, 43(2): 357-372. DOI:10.1016/0021-9991(81)90128-5

HUYNH H T. A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods[C]//18th AIAA Computational Fluid Dynamics Conference, Miami, Florida. Reston, Virigina: AIAA, 2007: 4079. doi: 10.2514/6.2007-4079

CASTONGUAY P, WILLIAMS D M, VINCENT P E, et al. Energy stable flux reconstruction schemes for advection-diffusion problems[J]. Computer Methods in Applied Mechanics and Engineering, 2013, 267: 400-417. DOI:10.1016/j.cma.2013.08.012

GUO Z L, ZHENG C G, SHI B C. Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method[J]. Chinese Physics, 2002, 11: 366-374. DOI: 10.1088/1009-1963/11/4/310

LEE T, LIN C L. An Eulerian description of the streaming process in the lattice Boltzmann equation[J]. Journal of Computational Physics, 2003, 185(2): 445-471. DOI:10.1016/S0021-9991(02)00065-7

GHIA U, GHIA K N, SHIN C T. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method[J]. Journal of Computational Physics, 1982, 48(3): 387-411. DOI:10.1016/0021-9991(82)90058-4

BERGER E, WILLE R. Periodic flow phenomena[J]. Annual Review of Fluid Mechanics, 1972, 4(1): 313-340. DOI:10.1146/annurev.fl.04.010172.001525

CHIU P H, LIN R K, SHEU T W H. A differentially interpolated direct forcing immersed boundary method for predicting incompressible Navier-Stokes equations in time-varying complex geometries[J]. Journal of Computational Physics, 2010, 229(12): 4476-4500. DOI:10.1016/j.jcp.2010.02.013

LI W D, LI W, SONG P, et al. A conservation-moment-based implicit finite volume lattice Boltzmann method for steady nearly incompressible flows[J]. Journal of Computational Physics, 2019, 398: 108882. DOI:10.1016/j.jcp.2019.108882

GHAFFARIAN A, HEJRANFAR K. An implicit dual-time stepping spectral difference lattice Boltzmann method for simulation of viscous compressible flows on structured meshes[J]. Meccanica, 2019, 54(10): 1561-1581. DOI:10.1007/s11012-019-01036-w

MA C, WU J, JIANG L. A weighted essentially nonoscillatory-based phase field lattice Boltzmann method for incompressible two-phase flows with high density contrast[J]. International Journal for Numerical Methods in Fluids, 2021, 93(7): 2272-2290. DOI:10.1002/fld.4973

MA C, WU J, ZHANG T W. A high order spectral difference-based phase field lattice Boltzmann method for incompressible two-phase flows[J]. Physics of Fluids, 2020, 32(12): 122113. DOI:10.1063/5.0033204

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