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旋转肥皂泡热对流能量耗散与边界层特性的数值模拟

贺啸秋 熊永亮 彭泽瑞 徐顺

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旋转肥皂泡热对流能量耗散与边界层特性的数值模拟

贺啸秋, 熊永亮, 彭泽瑞, 徐顺

Boundary layers and energy dissipation rates on a half soap bubble heated at the equator

He Xiao-Qiu, Xiong Yong-Liang, Peng Ze-Rui, Xu Shun
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  • 将底部加热的半个肥皂泡作为一个新的热对流模型, 结合了肥皂泡固有的球面与准二维特征, 由此有助于理解行星大气流动中的复杂物理机制与热对流特性. 本文使用直接数值模拟方法计算了旋转肥皂泡上的湍流热对流, 研究了肥皂泡上的温度与黏性边界层以及拟热能和动能耗散规律. 结合肥皂泡上温度场与速度场特征, 分别根据温度脉动均方根最大值以及速度脉动边界处斜率延长线与最大值交点提出了肥皂泡上温度与黏性边界层的识别方法. 研究发现, 当肥皂泡从边界吸收能量时, 拟热能耗散与动能耗散均集中在边界层中, 肥皂泡上的温度边界层与黏性边界层厚度与瑞利数$Ra$存在明确的标度关系. 相比经典Rayleigh-Bénrad对流(RB对流)模型, 温度标度指数具有较为接近的结果, 但速度标度指数存在一定的差异. 此外, 在混合区, 均方根温度($T^{*}$)随纬度($\theta$)具有近似$T^{*}\sim\theta^{0.5}$的标度关系, 这与RB对流模型及其相应的理论预测一致. 最后通过能量平衡方程发现, 肥皂泡上拟热能内耗散率$\varepsilon_{{T}}^0$和动能内耗散率$\varepsilon_{{u}}^0$比拟热能外耗散率$\varepsilon_{{T}}^1$和动能外耗散率$\varepsilon_{{u}}^1$大1个量级, 拟热能与动能的内部耗散率在边界层中具有支配地位, 随着肥皂泡旋转速率的增加, 热羽流难以输运到高纬度地区, 进一步降低了拟热能与动能外耗散率的影响.
    The soap bubble heated at the bottom is a novel thermal convection cell, which has the inherent spherical surface and quasi two-dimensional features, so that it can provide an insight into the complex physical mechanism of the planetary or atomspherical flows. This paper analyses the turbulent thermal convection on the soap bubble and addresses the properties including the thermal layer and the viscous boundary layer, the thermal dissipation and the kinetic dissipation by direct numerical simulation (DNS). The thermal dissipation and the kinetic dissipation are mostly occur in the boundary layers. They reveal the great significance of the boundary layers in the process of the energy absorption. By considering the complex characteristics of the heated bubble, this study proposes a new definition to identify the thermal boundary layer and viscous boundary layer. The thermal boundary layer thickness of $\delta_{T}$ is defined as the geodetic distance between the equator of the bubble and the latitude at which the the mean square root temperature ($T^{*}$) reaches a maximum value. On the other hand, the viscous boundary layer thickness $\delta_{u}$ is the geodetic distance from the equator at the latitude where the extrapolation for the linear part of the mean square root turbulent latitude velocity ($u^{*}_{\theta}$) meets its maximum value. It is found that $\delta_{T}$ and $\delta_{u}$ both have a power-law dependence on the Rayleigh number. For the bubble, the scaling coefficent of $\delta_{T}$ is $-0.32$ which is consistent with that from the Rayleigh-Bénard convection model. The rotation does not affect the scaling coefficent of $\delta_{T}$. On the other hand, the scaling coefficent of $\delta_{u}$ equals $-0.20$ and is different from that given by the Rayleigh-Bénard convection model. The weak rotation does not change the coefficent while the strong rotation makes it increase to $-0.14$. The profile of $T^{*}$ satisfies the scaling law of $T^{*}\sim\theta^{0.5}$ with the latitude of ($\theta$) on the bubble. The scaling law of the mean square root temperature profile coincides with the theoretical prediction and the results obtained from the Rayleigh-Bénard convection model. However, the strong rotation is capable of shifting the scaling coefficent of the power law away from $0.5$ and shorterning the interval of satisfying the power law. Finally, it is found that the internal thermal dissipation rate and kinetic dissipation rate $\varepsilon^0_T$ and $\varepsilon^0_u$ are one order larger than their peers: the external thermal dissipation and kinetic dissipation rates $\varepsilon^1_T$ and $\varepsilon^1_u$ based on a thorough analysis of the energy budget. The major thermal dissipation and kinetic dissipation are accumulated in the boundary layers. With the rotation rate increasing, less energy is transfered from the bottom to the top of the bubble and the influence of the external energy dissipations is less pronounced.
      通信作者: 熊永亮, xylcfd@hust.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 11872187, 12072125)资助的课题
      Corresponding author: Xiong Yong-Liang, xylcfd@hust.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11872187, 12072125)
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  • 图 1  不同坐标系下的计算域与球极投影平面投影示意图

    Fig. 1.  Computational domain under different coordinate system and the illustration of the stereographical projection

    图 2  网格结点分布图

    Fig. 2.  Node distribution inside a grid cell

    图 3  不同$1/Ro$$Ra$下, T在肥皂泡上的分布 $({\rm{a}})$ 算例$Ra=3\times10^6$, $1/Ro=0$; $({\rm{b}})$ 算例$Ra=3\times10^9$, $1/Ro=0$; $({\rm{c}})$ 算例$Ra=3\times10^9$, $1/Ro=10$

    Fig. 3.  The instantanous temperature field with different $1/Ro$ and $Ra$: $({\rm{a}})$ The case of $Ra=3\times10^6$, $1/Ro=0$; $({\rm{b}})$ the case of $Ra=3\times10^9$, $1/Ro=0$; $({\rm{c}})$ the case of $Ra=3\times10^9$, $1/Ro=10$.

    图 4  不同$1/Ro$$Ra$下, $\log{E_k}$在肥皂泡上的分布 $({\rm{a}})$ 算例$Ra=3\times10^6$, $1/Ro=0$; $({\rm{b}})$ 算例$Ra=3\times10^9$, $1/Ro=0$; $({\rm{c}})$ 算例$Ra=3\times10^9$, $1/Ro=10$

    Fig. 4.  The instantanous filed of $\log{E_k}$ with different $1/Ro$ and $Ra$: $({\rm{a}})$The case of $Ra=3\times10^6$, $1/Ro=0$; $({\rm{b}})$ the case of $Ra=3\times10^9$, $1/Ro=0$; $({\rm{c}})$ the case of $Ra=3\times10^9$, $1/Ro=10$.

    图 5  不同$1/Ro$$Ra$下, 平均温度$\langle T\rangle$随纬度$\theta$的变化规律 $({\rm{a}})$ 算例$Ra=3\times10^6$; $({\rm{b}})$ 算例$Ra=3\times10^7$; $({\rm{c}})$ 算例$Ra=3\times $$ 10^8$; $({\rm{d}})$ 算例$Ra=3\times10^9$

    Fig. 5.  The variation of mean temperature $\langle T\rangle$ with the latitude $\theta$ for the different $1/Ro$ and $Ra$: $({\rm{a}})$ The cases of $Ra=3\times10^6$; $({\rm{b}})$ the cases of $Ra=3\times10^7$; $({\rm{c}})$ the cases of $Ra=3\times10^8$; $({\rm{d}})$ the cases of $Ra=3\times10^9$

    图 6  $Nu$(上)和$Re$(下)随$Ra$(左)和$1/Ro$(右)的变化规律 (a) 在$1/Ro$恒定的条件下, $Nu$$Ra$的标度规律; (b) 在$Ra$恒定的条件下, $Nu$$1/Ro$的变化规律; (c) 在$1/Ro$恒定的条件下, $Re$$Ra$的标度规律; (d) 在$Ra$恒定的条件下, $Re$$Ra$的变化规律

    Fig. 6.  The variation of $Nu$(up) and $Re$(down) with $Ra$(left) and $1/Ro$(right): (a) The scaling behavior of $Nu$ with $Ra$ in condition of fixed $1/Ro$; (b) the variation of $Nu$ with $1/Ro$ in condition of fixed $Ra$; (c) the scaling behavior of $Re$ with $Ra$ in condition of fixed $1/Ro$; (d) the variation of $Re$ with $1/Ro$ in condition of fixed $Ra$

    图 7  (a)均方根温度剖面分布与温度边界层定义示意图; (b)不同$1/Ro$下, 温度边界层厚度随$Ra$变化规律

    Fig. 7.  (a) The variation of $T^{*}$ profile with $Ra$; (b) the thickness of the thermal boundary layers with $Ra$ and $1/Ro$

    图 8  不同$1/Ro$$Ra$条件下, 使用$\delta_{T}$$\max(T^{*})$进行归一化之后的$T^{*}$曲线 (a) $1/Ro=0$; (b) $1/Ro=0.1$; (c) $1/Ro=1$; (d) $1/Ro=10$

    Fig. 8.  The RMS temperature distribution normalized by $\delta_{T}$ and $\max(T^{*})$ for the different $1/Ro$ and $Ra$: (a) $1/Ro=0$; (b) $1/Ro=0.1$; (c) $1/Ro=1$; (d) $1/Ro=10$

    图 9  不同$1/Ro$$Ra$条件下, $T^{*}$的剖面曲线 (a) $1/Ro=0$; (b) $1/Ro=0.1$; (c) $1/Ro=1$; (d) $1/Ro=10$

    Fig. 9.  The profile of $T^{*}$ for the different $1/Ro$ and $Ra$: (a) $1/Ro=0$; (b) $1/Ro=0.1$; (c) $1/Ro=1$; (d) $1/Ro=10$

    图 10  不同$Ra$$1/Ro$条件下, 均方根纬度速度$u^{*}_{\theta}$纬度剖面曲线 (a) $Ra=3\times10^6$; (b) $Ra=3\times10^7$; (c) $Ra=3\times10^8$; (d) $Ra=3\times10^9$

    Fig. 10.  The profile of the RMS velocity in the latitude direction $u^{*}_{\theta}$ for the differnet $Ra$ and $1/Ro$: (a) $Ra=3\times10^6$; (b) $Ra= $$ 3\times10^7$; (c) $Ra=3\times10^8$; (d) $Ra=3\times10^9$

    图 11  (a)黏性边界层厚度$\delta_u$的定义方法(示意算例$Ra=3\times10^7$, $1/Ro=0$); (b)不同$1/Ro$$\delta_u$$Ra$变化规律

    Fig. 11.  (a) The definition of the viscous boundary layer thicknesses $\delta_u$ with the example case of $Ra=3\times10^7$ and $1/Ro=0$; (b) the variation of $\delta_u$ with $Ra$ for the different $1/Ro$

    图 12  不同$Ra$$1/Ro$条件下, 拟热能内耗散率$\varepsilon_{T}^{0}$在纬度方向的分布 (a) $Ra=3\times10^6$; (b) $Ra=3\times10^7$; (c) $Ra=3\times $$ 10^8$; (d) $Ra=3\times10^9$. 子图为边界层附近的放大

    Fig. 12.  The distribution of the internal thermal energy dissipation rate $\varepsilon_{T}^{0}$ in the latitude direction for the different $1/Ro$ and $Ra$: (a) $Ra=3\times10^6$; (b) $Ra=3\times10^7$; (c) $Ra=3\times10^8$; (d) $Ra=3\times10^9$. The insets are the zoom-in for the boundary layers

    图 13  不同$Ra$$1/Ro$条件下, 拟热能内耗散率$\varepsilon_{T}^{1}$在纬度方向的分布 (a) $Ra=3\times10^6$; (b) $Ra=3\times10^7$; (c) $Ra=3\times $$ 10^8$; (d) $Ra=3\times10^9$. 内插图为边界层附近的放大

    Fig. 13.  The distribution of the internal thermal energy dissipation rate $\varepsilon_{T}^{1}$ in the latitude direction for the different $1/Ro$ and $Ra$: (a) $Ra=3\times10^6$; (b) $Ra=3\times10^7$; (c) $Ra=3\times10^8$; (d) $Ra=3\times10^9$. The insets are the zoom-in for the boundary layers

    图 14  不同$Ra$$1/Ro$条件下, 动能耗散率以及浮力项在纬度方向的分布 (a), (b), (c) $Ra=3\times10^6$; (d), (e), (f) $Ra=3\times $$ 10^7$; (g), (h), (i) $Ra=3\times10^8$; (j), (k), (l) $Ra=3\times10^9$. 内插图为边界层附近的放大

    Fig. 14.  The distribution of the kinetic energy dissipation rate and the buoyancy term in the latitude direction for the different $1/Ro$ and $Ra$: (a), (b), (c) $Ra=3\times10^6$; (d), (e), (f) $Ra=3\times10^7$; (g), (h), (i) $Ra=3\times10^8$; (j), (k), (l) $Ra=3\times10^9$. The insets are the zoom-in for the boundary layers

    表 1  算例详细信息

    Table 1.  The detailed information of the cases

    $Ra$$1/Ro$$Pr$SFMesh resolution
    $3\times10^6$$0;0.1;1;10$70.060.06$1024\times1024$
    $3\times10^7$$0;0.1;1;10$ $1024\times1024$
    $3\times10^8$$0;0.1;1;10$ $1536\times1536$
    $3\times10^9$$0;0.1;1;10$$2048\times2048$
    下载: 导出CSV
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    Meuel T, Prado G, Seychelles F, Bessafi M, Kellay H 2012 Sci. Rep. 2 446Google Scholar

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    Meuel T, Coudert M, Fischer P, Bruneau C H, Kellay H 2018 Sci. Rep. 8 16513Google Scholar

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    He X Q, Bragg A D, Xiong Y L, Fischer P 2021 J. Fluid Mech. 924 A19Google Scholar

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    周全, 夏克青 2012 力学进展 42 231

    Zhou Q, Xia K Q 2012 Advances in Mechanics 42 231

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    谢毅超, 张路, 丁广裕, 陈鑫, 郗恒东, 夏克青 2022 力学进展

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出版历程
  • 收稿日期:  2022-04-14
  • 修回日期:  2022-08-20
  • 上网日期:  2022-10-05
  • 刊出日期:  2022-10-20

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