用改进的Lattice Boltzmann模型研究对流Cahn-Hilliard系统振荡

doi:10.16088/j.issn.1001-6600.2019.02.003

摘要/Abstract

摘要： 对流Cahn-Hilliard(简称C-H)系统是一类连续介质模型,可描述二元系统中相变行为,其应用涉及固体理论、激发数学、材料科学等多个学科领域,且因其高度非线性的特征、丰富的动力学现象受到研究者广泛关注。本文利用Lattice Boltzmann (简称LB)方法将一维对流C-H系统的高阶空间导数项构造成源项,建立改进的LB方法计算格式,导出系统传播矩阵并分析了数值求解的稳定性对时空比例因子的选择要求。利用所得LB模型发现,系统对流强度k变大一方面会减小系统的振荡幅度A,并在较强对流时与系统振动幅度呈现出不依赖于初态的幂律特征;另一方面,系统的对流强度增加会促使系统的振荡频率变大,并在此过程中存在明显的频率跳变现象。进一步探究发现,不同初态可使系统演化至不同频率的周期振荡,而相同参数条件下多频共存的情况揭示了系统的动力学在大尺度下的混沌行为。

关键词: Lattice Boltzamann方法, 对流Cahn-Hilliard方程, 振荡幅度, 幂律特征, 频率跳变, 多频共存, 混沌

Abstract: Convection Cahn-Hilliard (abbreviated as C-H) system is a class of continuum medium models,which can be used to explore phase transitions in binary systems. Applications of C-H system have been so ubiquitous in solid theory,excitable mathematics,material science and so on, therefore, great attentions have been paid by researchers in recent years from the characteristics of high nonlinearity,rich and complicated dynamical phenomena. In this paper,it is proposed that improved Lattice Boltzmann (LB for short) method with source term from the spatial derivatives of high order to simulate 1D-convetion C-H system. According to such LB model,transfer matrix of system and the stable condition of numerical computation are explored,which are closely related to the ratio of lattice spacing and time step. Based on such LB model,interesting research results are realized:on one hand,the growing convection strength k decreases oscillation range of system A,and power law feature of k-A relationship is found independent of diverse initial states. On the other hand,although the growth of convection strength can promote oscillation frequency of system,frequency hopping phenomena are outstanding. Furthermore,different initial states can lead to periodic oscillations with different frequencies,which mean that oscillations with different frequencies can coexist in the same convection strength.

Key words: Lattice Boltzmann method, convection Cahn-Hilliard equation, oscillation range, power law, frequency hopping, frequency coexistence, chaos

中图分类号:

O411.3

张立升, 张智勇, 马凯华, 李国放. 用改进的Lattice Boltzmann模型研究对流Cahn-Hilliard系统振荡[J]. 广西师范大学学报（自然科学版）, 2019, 37(2): 15-26.

ZHANG Lisheng, ZHANG Zhiyong, MA Kaihua, LI Guofang. Studying Oscillations in Convection Cahn-Hilliard System with Improved Lattice Boltzmann Model[J]. Journal of Guangxi Normal University(Natural Science Edition), 2019, 37(2): 15-26.

参考文献

[1] CAHN J W. On spinodal decomposition[J]. Acta Metall,1961,9(3):795-501.
[2] CAHN J W,HILLIARD J E. Free energy of a non-uniform system I. Interfacial free energy[J]. J Chem Phys,1958,2(2):258-267.
[3] 李德生,李清仪. Cahn-Hilliard方程的动力学稳定性[J]. 数学学报,2000,43(1):127-134.
[4] 曹保胜,何洋洋,董斌. 纳米尺度下体系自由能对扩散模型的影响[J]. 材料导报,2009,23(11):113-116.
[5] GOLOVIN A A,NEPOMNYASHCHY A A,DAVIS S H,et al. Convective Cahn-Hilliard models:from coarsening to roughening[J]. Phys Rev Lett,2001,86(8):1550-1553.
[6] 罗勇,欧阳文斌,杨其,等. 振荡剪切场下PS/PVME共混物的相分离动力学研究[J]. 高分子学报,2006,4(4):557-563.
[7] 赵才地. 非线性Cahn-Hilliard方程的行波解[J]. 武汉科技大学学报(自然科学版),2006,29(2):215-216.
[8] GUO Xiulan,LI Kaitai,YANG Shouzhi. The global attractor for a sort of fourth order nonlinear parabolic equations[J]. Mathematica Applicata,2002,15(3):1-7.
[9] 张正丽,张强. 一类Cahn-Hilliard方程的定态分歧[J]. 四川大学学报(自然科学版),2011,48(4):729-732.
[10] 曹保胜,张志鹏,雷明凯. 二元非均匀体系非线性动力学扩散模型的相关性[J]. 金属学报,2008,44:281.
[11] SAVINA T V,GOLOVIN A A,DAVIS S H,et al. Faceting of a growing crystal surface by surface diffusion[J]. Phys Rev E,2003,67:021606.
[12] MIKHAIL K. Long-wave model for strongly anisotropic growth of a crystal step[J]. Phys Rev E,2013,88:022402.
[13] 梁宏,柴振华,施保昌. 分叉微通道内液滴动力学行为的格子Boltzmann 方法模拟[J]. 物理学报,2016,65(20):204701.
[14] 上官勇刚,吴远志,陈锋,等. 制备方法对聚合物共混物相分离与黏弹弛豫的影响[J]. 中国科学:化学,2011,41(2):391-397.
[15] 黄锐. Cahn-Hilliard方程的径向对称稳态解[D]. 长春:吉林大学,2004.
[16] 张铁. Cahn-Hilliard方程有限元分析[J]. 计算数学,2006,28(3):281-292.
[17] JELI$\acute{C}$ A,ILG P,ÖTTINGER H C. Bridging length and time scales in sheared demixing systems:From the Cahn-Hilliard to the Doi-Ohta model[J]. Phys Rev E,2010,81:011131.
[18] MITLIN V. On the μ-transform:Applications to the Cahn-Hilliard equation[J]. Phys Lett A,2006,58:142-148.
[19] LI Juan,SUN Zhizhong,ZHAO Xuan. A three level linearized compact difference scheme for the Cahn-Hilliard equation[J]. Science China:Mathematics,2012,55(4):805-826.
[20] GRUNAU D,CHEN S,EGGERT K A. Lattice Boltzmann model for multiphase fluid flows[J]. Phys Fluids,1993,A5:2557.
[21] 许爱国,张广财,甘延标. 相分离过程的离散Boltzmnn方法研究进展[J]. 力学与实践,2016,38(4):361-374.
[22] LAI H L,MA C F. Lattice Boltzmann method for the generalized Kuramoto-Sivashinsky equation[J]. Physica A,2009,388:1405-1412.
[23] CHAI Z H,HE N Z,GUO Z L,et al. Lattice Boltzmann model for high-order nonlinear partial differential equations[J]. Phys Rev E,2018,97:013304.
[24] McNAMARA G,ZANETTI G. Use of the Boltzmann equation to simulation lattice gas automata[J]. Phys Rev Lett,1988,61(20/22):2332-2335.
[25] CHEN S Y,CHEN H D,MARITINEZ D,et al. Lattice Boltzmann model simulation of magneto hydynamics[J]. Phys Rev Lett,1991,67(27):3776-3780.
[26] 王勇. 格子Boltzmann方法在热声领域的应用及热声谐振管可视化实验研究[D]. 西安:西安交通大学,2009.
[27] 陈杰,钱跃竑. 热格子Boltzmann方法分析及应用[J]. 上海大学学报(自然科学版),2012,18(5):489-495.
[28] 邓敏艺,刘慕仁,何云,等. 基于九速四方格子模型的二维对流扩散方程格子Boltzmann 方法模拟[J]. 计算物理,2000,17(1/2):161-165.
[29] QIAN Y H,SUCCI S,ORSZAG S A,Recent advances in Lattice Boltzmann computing[J]. Annual Reviews of Computational Physics,1995,3:195-242.
[30] 史秀波. 用于波动方程的格子Boltzmann方法及数值模拟研究[D]. 长春:吉林大学,2010.
[31] CHAI Z H,SHI B C,ZHENG L. A unified Lattice Boltzmann model for some nonlinear partial differential equations[J]. Chaos,Solitons and Fractals,2008,36:874-882.
[32] BÖSCH F,KARLIN I V. Exact Lattice Boltzmann equation[J]. Phys Rev Lett,2013,111:090601.
[33] XU A G,ZHANG G C,GAN Y B,et al. Lattice Boltzmann modeling and simulation of compressible flows[J]. Front Phys,2012,7(5):582.
[34] MENDOZA M,BOGHOSIAN B M,HERRMANN H J,et al,Fast Lattice Boltzmann solver for relativistic hydrodynamics[J]. Phys Rev Lett,2010,105(1):014502.
[35] LIN Chuandong,LUO Kaihong,FEI Linlin,et al. A multi-component discrete Boltzmann model for nonequilibrium reactive flows[J]. Sci Rep,2017,7:14580.
[36] LIANG H,SHI B C,GUO Z L,et al. Phase-field-based multiple-relaxation-time Lattice Boltzmann model for incompressible multiphase flows[J]. Phys Rev E,2014,89:053320.
[37] ZHENG Lin,ZHENG Song,ZHAI Qinglan. Lattice Boltzmann equation method for the Cahn-Hilliard equation[J]. Phys Rev E,2015,91:013309.
[38] 许爱国,张广财,李英骏,等. 非平衡与多相复杂系统模拟研究:Lattice Boltzmann 动力学理论与应用[J]. 物理学进展,2014,24(3):136-167.

Metrics

Viewed

Full text

Abstract

Cited

Shared

Discussed