二维湍流热对流最大速度Re数特性及流态突变特征Re数

何建超; 方明卫; 包芸

doi:10.7498/aps.71.20220352

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摘要
本文计算系列二维湍流热对流，Prandtl (Pr)数从0.25到100，Rayleigh (Ra)数从1×10⁷到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≈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_c Pr^-1.5=10⁹。

关键词:
热对流 /

Reynolds数 /

流态 /

Prandtl数
Abstract
The Ra dependence of Ra in Rayleigh-Bénard(RB) convection has been reported by many studies, but the power-law scaling is different in these works. Previous studies have found that when Ra reaches the critical, the flow patterns change and a transition appears in the scaling of Nu(Ra) and Re(Ra). The Grossmann-Lohse(GL) model divides the Ra-Pr Phase into several regions to predict the scaling of Nu(Ra,Pr) and Re(Ra,Pr), indicating that the thermal dissipation and kinetic dissipation behaviors are diverse in the different region. 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 paper, we conduct a series of numerical simulations in 2-D 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 happen when Ra reaches a critical value Ra_c, and Ra_c increases with larger Pr. The Re number, which is defined with the maximum velocity, also shows a plateau at Ra_c. Before and after Ra_c, the Ra scaling exponents of Re remain 0.55, which get smaller at very high Ra. Specially, under different Pr, 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 by Pr^-1.5 to disscuss the relationship between Re and RaPr^-1.5. It is interesting that the Re(RaPr^-1.5) at different Pr 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 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 found that the large scale circulation(LSC) changes from ellipse to circle at Ra_c. In other words, the flow patterns will change to circular LSC at Re_c at different Pr, and Re_c is a constant as mention above. This finding could 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.

Keywords:
Thermal convection /

Reynolds number /

Flow pattern /

Prandtl number
施引文献

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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

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)

摘要: 本文计算系列二维湍流热对流，Prandtl (Pr)数从0.25到100，Rayleigh (Ra)数从1×10⁷到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≈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_c Pr^-1.5=10⁹。

关键词:

热对流 /
Reynolds数 /
流态 /
Prandtl数

The scaling of Reynolds number based on the maximum velocity and characteristic Reynolds number in two-dimensional thermal turbulence convection

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

Abstract: The Ra dependence of Ra in Rayleigh-Bénard(RB) convection has been reported by many studies, but the power-law scaling is different in these works. Previous studies have found that when Ra reaches the critical, the flow patterns change and a transition appears in the scaling of Nu(Ra) and Re(Ra). The Grossmann-Lohse(GL) model divides the Ra-Pr Phase into several regions to predict the scaling of Nu(Ra,Pr) and Re(Ra,Pr), indicating that the thermal dissipation and kinetic dissipation behaviors are diverse in the different region. 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 paper, we conduct a series of numerical simulations in 2-D 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 happen when Ra reaches a critical value Ra_c, and Ra_c increases with larger Pr. The Re number, which is defined with the maximum velocity, also shows a plateau at Ra_c. Before and after Ra_c, the Ra scaling exponents of Re remain 0.55, which get smaller at very high Ra. Specially, under different Pr, 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 by Pr^-1.5 to disscuss the relationship between Re and RaPr^-1.5. It is interesting that the Re(RaPr^-1.5) at different Pr 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 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 found that the large scale circulation(LSC) changes from ellipse to circle at Ra_c. In other words, the flow patterns will change to circular LSC at Re_c at different Pr, and Re_c is a constant as mention above. This finding could 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.

Keywords:

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