二维湍流热对流最大速度<i>Re</i>数特性及流态突变特征<i>Re</i>数

何建超; 方明卫; 包芸

doi:10.7498/aps.71.20220352

摘要

本文计算系列二维湍流热对流, Prandtl(Pr)数和Rayleigh(Ra)数范围分别为0.25—100和1×10⁷—1×10¹², 研究Reynolds(Re)数的变化规律. 以最大速度计算的Re数与Ra数存在标度律关系, 但中间出现间断. 研究表明, 大尺度环流形态由椭圆形到圆形的突变引起流动失稳, 导致最大速度值间断下降, 影响Re数变化趋势的连续性. 所有Pr数对应的流态突变特征Re数为常值, Re_c约为1.4 × 10⁴, 即当Re数达到特征Re_c时, 大尺度环流形态会发生从椭圆形到圆形的突变. 间断点对应的Ra_c与Pr数之间存在标度关系Ra_c-Pr^1.5. 对Ra数进行补偿平移, 所有Pr数的Re与RaPr^–1.5的变化曲线重合, 不同Pr数有相同的间断临界点位置, Ra_cPr^–1.5 = 10⁹.

关键词:

热对流 /
Reynolds数 /
流态 /
Prandtl数

Abstract

Rayleigh number (Ra) dependence in Rayleigh-Bénard (RB) convection has been studied by many investigators, but the reported power-law scaling expressions are different in these researches. Previous studies have found that when Ra reaches a critical value, the flow patterns change and a transition appears in the scaling of Nu(Ra) (where Nu represents Nusselt number) and Re(Ra) (where Re denotes Reynold number). The Grossmann-Lohse(GL) model divides the Ra-Pr(where Pr refers to Prandtl number) phase into several regions to predict the scaling expressions of Nu(Ra,Pr) and Re(Ra,Pr), indicating that the thermal dissipation behavior and kinetic dissipation behaviors are diverse in the different regions. Moreover, some physical quantities also show a transition and some structures in the flow fields, such as large scale circulation and boundary layer, change when Ra increases. In this work, we conduct a series of numerical simulations in two-dimensional RB convection with Ra ranging from 10⁷ to 10¹² and Pr ranging from 0.25 to 100, which is unprecedentedly wide. The relationship between the maximum velocity and Ra is investigated, and an unexpected drop happens when Ra reaches a critical value Ra_c, and Ra_c increases with Pr increasing. The Re number, which is defined as a maximum velocity, also shows a plateau at Ra_c. Before and after Ra_c, the Ra scaling exponent of Re remains 0.55, which gets smaller at very high Ra. Specially, under different Pr values, the plateau appears at Re_c ≈ 1.4 × 10⁴. In addition, a scaling Ra_c~Pr^1.5 is found and the Ra is compensated for by Pr^–1.5 to disscuss the relationship between Re and RaPr^–1.5. It is interesting that the Re(RaPr^–1.5) expressons at different Pr values well coincide, indicating a self-similarity of Re(RaPr^–1.5). The plateau appears at RaPr^–1.5 = 1 × 10⁹, meaning that Re_c would reach 1.4 × 10⁴ at any Pr value when RaPr^–1.5 = 1 × 10⁹. To further investigate the plateau of Re, the flow patterns are compared with time-averaged velocity fields and we find that the large scale circulation (LSC) changes from ellipse to circle at Ra_c. In other words, the flow pattern will change into circular LSC at Re_c at different Pr values, and Re_c is a constant as mentioned above. This finding can help us to distinguish the two flow patterns with given Ra and Pr, and to predict the Re scaling in an appropriate range of Ra with different Pr values.

Keywords:

作者及机构信息

1.
北京航空航天大学, 航空科学与工程学院, 北京　100191

2.
中山大学, 航空航天学院, 深圳　518107

通信作者: 包芸, stsby@mail.sysu.edu.cn

基金项目: 国家自然科学基金(批准号: 11772362)资助的课题.

Authors and contacts

1.
School of Aeronautics Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China

2.
School of Aeronautics and Astronautics, Sun Yat-Sen University, Shenzhen 518107, China

Corresponding author: Bao Yun, stsby@mail.sysu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11772362 ).

文章全文

参考文献

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施引文献

图 1 二维热对流计算算例Ra-Pr相图

Fig. 1. The Ra-Pr Phase diagram explored in the present study.

下载: 全尺寸图片幻灯片

图 2 平均速度分布及最大速度点

Fig. 2. The time-averaged velocity filed and the maximum velocity point.

下载: 全尺寸图片幻灯片

图 3 Re数与Ra数关系

Fig. 3. Re as a function of Ra

下载: 全尺寸图片幻灯片

图 4 平均速度场变化及流态突变(Pr = 0.7)　(a) Ra = 2 × 10⁸; (b) Ra = 5 × 10⁸; (c) Ra = 1 × 10⁹; (d) Ra = 2 × 10⁹

Fig. 4. The time-averaged velocity fieldsc and the sudden change of flow pattern (Pr = 0.7): (a) Ra = 2 × 10⁸; (b) Ra = 5 × 10⁸; (c) Ra = 1 × 10⁹; (d) Ra = 2 × 10⁹.

下载: 全尺寸图片幻灯片

图 5 最大速度随Ra数变化

Fig. 5. The maximum velocity as a function of Ra

下载: 全尺寸图片幻灯片

图 6 Re数与Ra的关系图

Fig. 6. Re as a function of Ra

下载: 全尺寸图片幻灯片

图 7 不同Pr数下Re数与Ra数的关系

Fig. 7. Re as a function of Ra at different Pr

下载: 全尺寸图片幻灯片

图 8 不同Pr数下Re数与RaPr^–1.5的关系

Fig. 8. Re as a function of RaPr^–1.5 at different Pr

下载: 全尺寸图片幻灯片

[1]	Malkus W V R 1954 Proc. R. Soc. Lond. A 225 196 Google Scholar
[2]	Globe S, Dropkin D 1959 J. Heat Transfer 81 24 Google Scholar
[3]	Shraiman B I, Siggia E D 1990 Phys. Rev. A 42 3650 Google Scholar
[4]	Grossmann S, Lohse D 2000 J. Fluid Mech. 407 27 Google Scholar
[5]	Grossmann S, Lohse D 2001 Phys. Rev. Lett. 86 3316 Google Scholar
[6]	Grossmann S, Lohse D 2002 Phys. Rev. E 66 016305 Google Scholar
[7]	Grossmann S, Lohse D 2004 Phys. Fluids 16 4462 Google Scholar
[8]	Stevens R J A M, Van Der Poel E P, Grossmann S, Lohse D 2013 J. Fluid Mech. 730 295 Google Scholar
[9]	Xia K Q, Lam S, Zhou S Q 2002 Phys. Rev. Lett. 88 064501 Google Scholar
[10]	van der Poel E P, Stevens R J A M, Lohse D 2013 J. Fluid Mech. 736 177 Google Scholar
[11]	Khurana A 1988 Phys. Today 41 17 Google Scholar
[12]	Willis G E, Deardorff J W 1970 J. Fluid Mech. 44 661 Google Scholar
[13]	Krishnamurti R 1970 J. Fluid Mech. 42 309 Google Scholar
[14]	Ahlers G 1974 Phys. Rev. Lett. 33 1185 Google Scholar
[15]	McLaughlin J B, Martin P C 1974 Phys. Rev. Lett. 33 1189 Google Scholar
[16]	Heslot F, Castaing B, Libchaber A 1987 Phys. Rev. A 36 5870 Google Scholar
[17]	Castaing B, Gunaratne G, Heslot F, Kadanoff L, Libchaber A, Thomae S, Wu X Z, Zaleski S, Zanetti G 1989 J. Fluid Mech. 204 1 Google Scholar
[18]	Sano M, Wu X Z, Libchaber A 1989 Phys. Rev. A 40 6421 Google Scholar
[19]	包芸, 何建超, 高振源 2019 物理学报 68 164701 Google Scholar Bao Y, He J C, Gao Z Y 2019 Acta Phys. Sin. 68 164701 Google Scholar
[20]	Gao Z Y, Luo J H, Bao Y 2018 Chin. Phys. B 27 104702 Google Scholar
[21]	Verzicco R, Camuss I R 1999 J. Fluid Mech. 383 55 Google Scholar
[22]	Kerr R M 1996 J. Fluid Mech. 310 139 Google Scholar
[23]	Li X M, He J D, Tian Y, Hao P, Huang S D 2021 J. Fluid Mech. 915 A60 Google Scholar
[24]	Chen X, Huang S D, Xia K Q, Xi H D 2019 J. Fluid Mech. 877 R1 Google Scholar
[25]	Xu A, Chen X, Xi H D 2021 Phys. Fluids 910 A33 Google Scholar
[26]	Werne J, Deluca EE, Rosner R, Cattaneo F 1991 Phys. Rev. Lett. 67 3919 Google Scholar
[27]	Bao Y, Luo J H, Ye M X 2017 J. Mech. 34 159 Google Scholar
[28]	Sun X H 1995 Parallel Comput. 21 1241 Google Scholar
[29]	Shishkina O, Stevens R J A M, Grossmann S, Lohse D 2010 New J. Phys. 12 075022 Google Scholar
[30]	He J C, Fang M W, Gao Z Y, Huang S D, Bao Y 2021 Chin. Phys. B 30 094701 Google Scholar
[31]	Gao Z Y, Bao Y, Huang S D 2019 72nd Annual Meeting of the APS Division of Fluid Dynamics Seattle, Washington, November 23–26, G14.00003

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二维湍流热对流最大速度Re数特性及流态突变特征Re数