搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

弱Soret效应混合流体对流系统的分岔与非线性演化

郑来运 赵秉新 杨建青

引用本文:
Citation:

弱Soret效应混合流体对流系统的分岔与非线性演化

郑来运, 赵秉新, 杨建青

Bifurcation and nonlinear evolution of convection in binary fluid mixtures with weak Soret effect

Zheng Lai-Yun, Zhao Bing-Xin, Yang Jian-Qing
PDF
HTML
导出引用
  • 混合流体Rayleigh-Bénard(RB)对流是研究非平衡耗散系统的自组织斑图及非线性动力学特性的典型模型. 本文利用高精度数值方法模拟了底部均匀加热的矩形腔体中混合流体RB对流, 研究了具有极微弱Soret效应(分离比$\psi=-0.02$)的混合流体对流的分岔特性及斑图的形成和演化, 给出了分岔曲线图. 获得了行波交替闪动的Blinking状态、局部行波对流和定常对流(SOC)三种稳定状态, 讨论了状态之间的过渡. 研究发现从Blinking状态到局部行波对流状态的过渡存在迟滞现象, 过渡时行波频率、对流振幅和对流传热Nusselt数等均有明显的跳跃. 在Blinking状态存在的Rayleigh数区间下界附近, 外部施加的不对称初始扰动是形成该状态的诱因. 随着Rayleigh数增大, 临界SOC状态经过多次分岔并形成多个具有不同波数的SOC状态后过渡为混沌状态.
    Rayleigh-Bénard (RB) convection in binary fluid mixtures, which shows rich and interesting pattern formation behavior, is a paradigm for understanding instabilities, bifurcations, self-organization with complex spatiotemporal behavior and turbulence, with many applications in atmospheric and environmental physics, astrophysics, and process technology. In this paper, by using a high-order compact finite difference method to solve the full hydrodynamic field equations, we study numerically the RB convection in binary fluid mixtures such as ethanol-water with a very weak Soret effect (separation ratio $\psi=-0.02$) in a rectangular container heated uniformly from below. The direct numerical simulations are conducted in the rectangular container with aspect ratio of $\varGamma=12$ and with four no-slip and impermeable boundaries, isothermal horizontal and perfectly insulated vertical boundaries. The bifurcation and the origin and evolution of pattern in RB convection for the considered physical parameters are studied, and the bifurcation diagram is presented. By performing two-dimensional simulations, we observe three stable states of Blinking state, localized traveling wave and stationary overturning convection (SOC) state, and discuss the transitions between them. The results show that there is a hysteresis in the transition from the Blinking state to the localized traveling wave state for the considered separation ratio, and the evolution of the oscillation frequency, convection amplitude and Nusselt number are discontinuous. Near the lower bound of the Rayleigh number range where the Blinking state exists, a asymmetric initial disturbance is the inducement for the formation of the Blinking state. Inside the range, its inducing effect is weakened, and the oscillatory instability becomes the main reason. It is further confirmed that reflections of lateral walls are responsible for the survival of the stable Blinking state. With the increase of the Rayleigh number, the critical SOC state undergoes multiple bifurcations and forms multiple SOC states with different wave numbers, and then transitions to a chaotic state. There are no stable undulation traveling wave states at both ends of the critical SOC branch.
      通信作者: 赵秉新, zhao_bx@nxu.edu.cn
    • 基金项目: 国家级-国家自然科学基金(11662016)
      Corresponding author: Zhao Bing-Xin, zhao_bx@nxu.edu.cn
    [1]

    Cross M C, Hohenberg P C 1993 Rev. Mod. Phys. 65 851Google Scholar

    [2]

    Moses E, Fineberg J, Steinberg V 1987 Phys. Rev. A 35 2757Google Scholar

    [3]

    Heinrichs R, Ahlers G, Cannell D S 1987 Phys. Rev. A 35 2761Google Scholar

    [4]

    Fineberg J, Moses E, Steinberg V 1988 Phys. Rev. Lett. 61 838Google Scholar

    [5]

    Kolodner P, Surko C M 1988 Phys. Rev. Lett. 61 842Google Scholar

    [6]

    Barten W, Lücke M, Kamps M, Schmitz R 1995 Phys. Rev. E 51 5636Google Scholar

    [7]

    Barten W, Lücke M, Kamps M 1991 Phys. Rev. Lett. 66 2621Google Scholar

    [8]

    Barten W, Lücke M, Kamps M, Schmitz R 1995 Phys. Rev. E 51 5662Google Scholar

    [9]

    Batiste O, Net M, Mercader I, Knobloch E 2001 Phys. Rev. Lett. 86 2309Google Scholar

    [10]

    Batiste O, Knobloch E 2005 Phys. Rev. Lett. 95 244501Google Scholar

    [11]

    Ning L Z 2006 Rayleigh-Bénard convection in a binary fluid mixture with and without lateral flow (Xi’an: Northwest A&F University Press) pp41–56

    [12]

    李国栋, 黄永念 2007 物理学报 56 4742Google Scholar

    Li G D, Huang Y N 2007 Acta Phys. Sin. 56 4742Google Scholar

    [13]

    宁利中, 齐昕, 周洋, 余荔 2009 物理学报 58 2528Google Scholar

    Ning L Z, Qi X, Zhou Y, Yu L 2009 Acta Phys. Sin. 58 2528Google Scholar

    [14]

    Mercader I, Batiste O, Alonso A, Knobloch E 2011 J. Fluid Mech. 667 586Google Scholar

    [15]

    Mercader I, Batiste O, Alonso A, Knobloch E 2013 J. Fluid Mech. 722 240Google Scholar

    [16]

    王涛, 田振夫, 葛永斌 2011 水动学研究与进展(A辑) 26 41

    Wang T, Tian Z F, Ge Y B 2011 Chin. J. Hydrodyn. 26 41

    [17]

    Watanabe T, Iima M, Nishiura Y 2012 J. Fluid Mech. 712 219Google Scholar

    [18]

    Taraut A V, Smorodin B L, Lücke M 2012 New J. Phys. 14 093055Google Scholar

    [19]

    赵秉新 2012 水动力学研究与进展(A辑) 27 264

    Zhao B X 2012 Chin. J. Hydrodyn. 27 264

    [20]

    Shevtsova V, Gaponenko Y A, Sechenyh V, Melnikov D E, Lyubimova T, Mialdun A 2015 J. Fluid Mech. 767 290Google Scholar

    [21]

    Lyubimova T, Zubova N, Shevtsova V 2018 Microgravity Sci. Tec. 31 1

    [22]

    Alonso A, Mercader I, Batiste O 2018 Phys. Rev. E 97 023108Google Scholar

    [23]

    Smorodin B L, Ishutov S M, Myznikova B I 2017 Microgravity Sci. Technol. 30 95

    [24]

    Zhao B X, Tian Z F 2015 Phys. Fluids 27 074102Google Scholar

    [25]

    Mercader I, Batiste O, Alonso A, Knobloch E 2019 Phys. Rev. E 99 023113Google Scholar

    [26]

    Lyubimova T, Zubova N 2017 Int. J. Heat Mass Transfer 106 1134Google Scholar

    [27]

    宁利中, 刘爽, 宁碧波, 袁喆, 王新宏, 田伟利, 渠亚伟 2018 水动力学研究与进展(A辑) 33 515

    Ning L Z, Liu S, Ning B B, Tian W L, Qu Y W 2018 Chin. J. Hydrodyn. 33 515

    [28]

    宁利中, 徐泊冰, 宁碧波, 袁喆, 田伟利 2019 水动力学研究与进展(A辑) 34 93

    Ning L Z, Xu B B, Ning B B, Yuan Z, Tian W L 2019 Chin. J. Hydrodyn. 34 93

    [29]

    宁利中, 余荔, 袁喆, 周洋 2009 中国科学: 物理学 力学 天文学 39 746

    Ning L Z, Yu L, Yuan Z, Zhou Y 2009 Sci. Sin.-Phys. Mech. Astron. 39 746

    [30]

    宁利中, 王娜, 袁喆, 李开继, 王卓运 2014 物理学报 63 104401Google Scholar

    Ning L Z, Wang N, Yuan Z, Li K J, Wang Z Y 2014 Acta Phys. Sin. 63 104401Google Scholar

    [31]

    Qin Q, Xia Z A, Tian Z F 2014 Int. J. Heat Mass Transfer 71 405Google Scholar

    [32]

    Tian Z F, Liang X, Yu P X 2011 Int. J. Numer. Meth. Een. 88 511Google Scholar

    [33]

    Strogatz S H 1994 Nonlinear Dynamics and Chaos: With Applications To Physics, Biology, Chemistry, and Engineering (New York: Perseus Books Publishing LLC) pp58–60

    [34]

    Dangelmayr G, Knobloch E, Wegelin M 1991 EPL-Europhys. Lett. 16 723Google Scholar

  • 图 1  对流模型示意图

    Fig. 1.  Sketch of the two-dimensional convection model

    图 2  $ \psi=-0.02 $时Nusselt数$ Nu-1 $关于Rayleigh数r的分岔曲线. SOC$ _n $表示具有n个涡卷的SOC状态

    Fig. 2.  (a) Bifurcation diagram for $ \psi=-0.02 $. (b) close-up view of the part of the bifurcation diagram delimited by the straight dashed lines depicted in (a). Where SOC$ _n $ represents the SOC solutions with n rolls

    图 3  $ r=1.015 $时 (a) 流场时空结构, (b) $ w(0.13\varGamma, 0.5) $的时间序列和(c)功率谱密度

    Fig. 3.  (a) Spatio-temporal structure, (b) the time series of $ w(0.13\varGamma, 0.5) $ and (c) power spectral density for $ r=1.015 $

    图 4  $ r=1.015 $时, 两个观测点处垂向速度w随时间的发展

    Fig. 4.  The time series of the vertical velocity w at two monitoring points (a) $ (0.13\varGamma, 0.5) $ and (b) $ (0.87\varGamma, 0.5) $ for $ r=1.015 $

    图 5  $ r=1.0171 $时, Blinking状态与LTW状态流场典型波形和浓度场的比较

    Fig. 5.  Comparison of the lateral profiles and concentration fields between the Blinking and LTW states at $ r=1.0171 $: (a) The lateral profile and (c) concentration field of the Blinking state; (b) The lateral profile and (d) concentration field of the LTW state

    图 6  (a)闪动频率$ \omega_1 $和(b)行波频率$ \omega_2 $随Rayleigh数的变化

    Fig. 6.  The variation of (a) blinking frequency $ \omega_1 $ and (b) oscillation frequency $ \omega_2 $ as a function of the Rayleigh number

    图 7  Blinking和LTW状态的$ Nu-1 $r的变化情况. (b)为(a)中虚线标注矩形区域的局部放大

    Fig. 7.  The variation of $ Nu-1 $ of the Blinking and LTW states as a function of r. (b) Close-up view of the part delimited by the straight dashed lines depicted in (a)

    图 8  LTW状态的流场结构

    Fig. 8.  Structures of the flow field of LTW state: (a) Spatio-temporal structure; (b) a large-scale concentration current; (c) a transient structure of the concentration field

    图 9  $ r=1.13 $时流场时空发展和典型时刻的瞬时结构

    Fig. 9.  The spatio-temporal development and transient structures of the flow field at typical times for $ r=1.13 $

    图 10  $ r=1.13 $时 (a)$ Nu-1 $M的变化及(b)观测点处垂向速度的时间序列

    Fig. 10.  The variation of (a) $ Nu-1 $, M, and (b) the vertical velocity at the monitoring points for $ r=1.13 $

    图 11  $ r=1.13 $时SOC$ _{12} $状态的流场结构

    Fig. 11.  The structure of flow field for the SOC$ _{12} $ state at $ r=1.13 $: (a) The lateral profile on the horizontal centerline of the cavity; (b) the streamlines and the structure of the associated temperature field; (c) the structure of the concentration field

    图 12  SOC$ _{12} $状态Nusselt数随Rayleigh数的变化

    Fig. 12.  The variation of $ Nu $ with r for the SOC$ _{12} $ state

    表 1  分离比$ \psi = -0.10 $和–0.02时, 各状态临界Rayleigh数的比较

    Table 1.  Comparison of critical Rayleigh numbers for each state, $ \psi = -0.10 $ and $ -0.02 $

    $ \psi $ $ r_{\rm c} $ $ r_{\rm {sn}}^{\rm {SOC}} $ $ r_{\rm {start}}^{\rm {BTW}} $ $ r_{\rm {start}}^{\rm {LTW}} $ $ r^* $
    –0.10 1.111 1.062 1.089 1.145
    –0.02 1.035 1.008 1.013 1.0172 1.022
    下载: 导出CSV
  • [1]

    Cross M C, Hohenberg P C 1993 Rev. Mod. Phys. 65 851Google Scholar

    [2]

    Moses E, Fineberg J, Steinberg V 1987 Phys. Rev. A 35 2757Google Scholar

    [3]

    Heinrichs R, Ahlers G, Cannell D S 1987 Phys. Rev. A 35 2761Google Scholar

    [4]

    Fineberg J, Moses E, Steinberg V 1988 Phys. Rev. Lett. 61 838Google Scholar

    [5]

    Kolodner P, Surko C M 1988 Phys. Rev. Lett. 61 842Google Scholar

    [6]

    Barten W, Lücke M, Kamps M, Schmitz R 1995 Phys. Rev. E 51 5636Google Scholar

    [7]

    Barten W, Lücke M, Kamps M 1991 Phys. Rev. Lett. 66 2621Google Scholar

    [8]

    Barten W, Lücke M, Kamps M, Schmitz R 1995 Phys. Rev. E 51 5662Google Scholar

    [9]

    Batiste O, Net M, Mercader I, Knobloch E 2001 Phys. Rev. Lett. 86 2309Google Scholar

    [10]

    Batiste O, Knobloch E 2005 Phys. Rev. Lett. 95 244501Google Scholar

    [11]

    Ning L Z 2006 Rayleigh-Bénard convection in a binary fluid mixture with and without lateral flow (Xi’an: Northwest A&F University Press) pp41–56

    [12]

    李国栋, 黄永念 2007 物理学报 56 4742Google Scholar

    Li G D, Huang Y N 2007 Acta Phys. Sin. 56 4742Google Scholar

    [13]

    宁利中, 齐昕, 周洋, 余荔 2009 物理学报 58 2528Google Scholar

    Ning L Z, Qi X, Zhou Y, Yu L 2009 Acta Phys. Sin. 58 2528Google Scholar

    [14]

    Mercader I, Batiste O, Alonso A, Knobloch E 2011 J. Fluid Mech. 667 586Google Scholar

    [15]

    Mercader I, Batiste O, Alonso A, Knobloch E 2013 J. Fluid Mech. 722 240Google Scholar

    [16]

    王涛, 田振夫, 葛永斌 2011 水动学研究与进展(A辑) 26 41

    Wang T, Tian Z F, Ge Y B 2011 Chin. J. Hydrodyn. 26 41

    [17]

    Watanabe T, Iima M, Nishiura Y 2012 J. Fluid Mech. 712 219Google Scholar

    [18]

    Taraut A V, Smorodin B L, Lücke M 2012 New J. Phys. 14 093055Google Scholar

    [19]

    赵秉新 2012 水动力学研究与进展(A辑) 27 264

    Zhao B X 2012 Chin. J. Hydrodyn. 27 264

    [20]

    Shevtsova V, Gaponenko Y A, Sechenyh V, Melnikov D E, Lyubimova T, Mialdun A 2015 J. Fluid Mech. 767 290Google Scholar

    [21]

    Lyubimova T, Zubova N, Shevtsova V 2018 Microgravity Sci. Tec. 31 1

    [22]

    Alonso A, Mercader I, Batiste O 2018 Phys. Rev. E 97 023108Google Scholar

    [23]

    Smorodin B L, Ishutov S M, Myznikova B I 2017 Microgravity Sci. Technol. 30 95

    [24]

    Zhao B X, Tian Z F 2015 Phys. Fluids 27 074102Google Scholar

    [25]

    Mercader I, Batiste O, Alonso A, Knobloch E 2019 Phys. Rev. E 99 023113Google Scholar

    [26]

    Lyubimova T, Zubova N 2017 Int. J. Heat Mass Transfer 106 1134Google Scholar

    [27]

    宁利中, 刘爽, 宁碧波, 袁喆, 王新宏, 田伟利, 渠亚伟 2018 水动力学研究与进展(A辑) 33 515

    Ning L Z, Liu S, Ning B B, Tian W L, Qu Y W 2018 Chin. J. Hydrodyn. 33 515

    [28]

    宁利中, 徐泊冰, 宁碧波, 袁喆, 田伟利 2019 水动力学研究与进展(A辑) 34 93

    Ning L Z, Xu B B, Ning B B, Yuan Z, Tian W L 2019 Chin. J. Hydrodyn. 34 93

    [29]

    宁利中, 余荔, 袁喆, 周洋 2009 中国科学: 物理学 力学 天文学 39 746

    Ning L Z, Yu L, Yuan Z, Zhou Y 2009 Sci. Sin.-Phys. Mech. Astron. 39 746

    [30]

    宁利中, 王娜, 袁喆, 李开继, 王卓运 2014 物理学报 63 104401Google Scholar

    Ning L Z, Wang N, Yuan Z, Li K J, Wang Z Y 2014 Acta Phys. Sin. 63 104401Google Scholar

    [31]

    Qin Q, Xia Z A, Tian Z F 2014 Int. J. Heat Mass Transfer 71 405Google Scholar

    [32]

    Tian Z F, Liang X, Yu P X 2011 Int. J. Numer. Meth. Een. 88 511Google Scholar

    [33]

    Strogatz S H 1994 Nonlinear Dynamics and Chaos: With Applications To Physics, Biology, Chemistry, and Engineering (New York: Perseus Books Publishing LLC) pp58–60

    [34]

    Dangelmayr G, Knobloch E, Wegelin M 1991 EPL-Europhys. Lett. 16 723Google Scholar

  • [1] 赵雅琪, 刘谋天, 赵勇, 段利霞. 耦合前包钦格复合体神经元中复杂混合簇放电的动力学. 物理学报, 2021, 70(12): 120501. doi: 10.7498/aps.70.20210093
    [2] 姜伊澜, 陆博, 张万芹, 古华光. 快自突触反馈诱发混合簇放电的反常变化及分岔机制. 物理学报, 2021, 70(17): 170501. doi: 10.7498/aps.70.20210208
    [3] 尹慧, 赵秉新. 倾角对方腔内热对流非线性演化与分岔的影响. 物理学报, 2021, 70(11): 114401. doi: 10.7498/aps.70.20201513
    [4] 林茂杰, 常健, 吴宇昊, 徐山森, 魏炳波. 电磁悬浮条件下液态Fe50Cu50合金的对流和凝固规律研究. 物理学报, 2017, 66(13): 136401. doi: 10.7498/aps.66.136401
    [5] 宁利中, 胡彪, 宁碧波, 田伟利. Poiseuille-Rayleigh-Bnard流动中对流斑图的分区和成长. 物理学报, 2016, 65(21): 214401. doi: 10.7498/aps.65.214401
    [6] 石兰芳, 朱敏, 周先春, 汪维刚, 莫嘉琪. 一类非线性发展方程孤立子行波解. 物理学报, 2014, 63(13): 130201. doi: 10.7498/aps.63.130201
    [7] 宁利中, 王娜, 袁喆, 李开继, 王卓运. 分离比对混合流体Rayleigh-Bénard对流解的影响. 物理学报, 2014, 63(10): 104401. doi: 10.7498/aps.63.104401
    [8] 黄晨, 陈龙, 毕勤胜, 江浩斌. 机动车协商模型与分岔特性研究. 物理学报, 2013, 62(21): 210507. doi: 10.7498/aps.62.210507
    [9] 姚熊亮, 叶曦, 张阿漫. 行波驱动下空泡在可压缩流场中的运动特性研究. 物理学报, 2013, 62(24): 244701. doi: 10.7498/aps.62.244701
    [10] 李群宏, 闫玉龙, 杨丹. 耦合电路系统的分岔研究. 物理学报, 2012, 61(20): 200505. doi: 10.7498/aps.61.200505
    [11] 冯朝文, 蔡理, 杨晓阔, 康强, 彭卫东, 柏鹏. 单电子晶体管与金属氧化物半导体混合电路构造的一维离散混沌系统研究. 物理学报, 2012, 61(8): 080503. doi: 10.7498/aps.61.080503
    [12] 石玉峰, 许庆彦, 柳百成. 对流作用下枝晶形貌演化的数值模拟和实验研究. 物理学报, 2011, 60(12): 126101. doi: 10.7498/aps.60.126101
    [13] 陈章耀, 毕勤胜. Jerk系统耦合的分岔和混沌行为. 物理学报, 2010, 59(11): 7669-7678. doi: 10.7498/aps.59.7669
    [14] 姜泽辉, 张峰, 郭波, 赵海发, 郑瑞华. 受振颗粒“毛细”系统中的对流与有序化. 物理学报, 2010, 59(8): 5581-5587. doi: 10.7498/aps.59.5581
    [15] 包伯成, 康祝圣, 许建平, 胡文. 含指数项广义平方映射的分岔和吸引子. 物理学报, 2009, 58(3): 1420-1431. doi: 10.7498/aps.58.1420
    [16] 龙文元, 吕冬兰, 夏春, 潘美满, 蔡启舟, 陈立亮. 强迫对流影响二元合金非等温凝固枝晶生长的相场法模拟. 物理学报, 2009, 58(11): 7802-7808. doi: 10.7498/aps.58.7802
    [17] 孙东科, 朱鸣芳, 杨朝蓉, 潘诗琰, 戴挺. 强制对流和自然对流作用下枝晶生长的数值模拟. 物理学报, 2009, 58(13): 285-S291. doi: 10.7498/aps.58.285
    [18] 宁利中, 齐昕, 余荔, 周洋. 混合流体Rayleigh-Benard行波对流中的缺陷结构. 物理学报, 2009, 58(4): 2528-2534. doi: 10.7498/aps.58.2528
    [19] 张 维, 周淑华, 任 勇, 山秀明. Turbo译码算法的分岔与控制. 物理学报, 2006, 55(2): 622-627. doi: 10.7498/aps.55.622
    [20] 郝建红, 丁 武. 行波管放大器中辐射场的极限环振荡和混沌. 物理学报, 2003, 52(4): 906-910. doi: 10.7498/aps.52.906
计量
  • 文章访问数:  5832
  • PDF下载量:  72
  • 被引次数: 0
出版历程
  • 收稿日期:  2019-12-04
  • 修回日期:  2019-12-27
  • 刊出日期:  2020-04-05

/

返回文章
返回