Boundary layers and energy dissipation rates on a half soap bubble heated at the equator
Authors: He Xiao-Qiu , Xiong Yong-Liang , Peng Ze-Rui , Xu Shun ,   School of Aerospace Engineering, Huazhong University of Science and Technology, Wuhan 430074, China,   Hubei Key Laboratory of Engineering Structural Analysis and Safety Assessment, Wuhan 430074, China
Published: September 2022

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 <inline-formula><tex-math id="M9">\begin{document}$\delta_{T}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M9.png"/></alternatives></inline-formula> is defined as the geodetic distance between the equator of the bubble and the latitude at which the the mean square root temperature (<inline-formula><tex-math id="M10">\begin{document}$T^{*}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M10.png"/></alternatives></inline-formula>) reaches a maximum value. On the other hand, the viscous boundary layer thickness <inline-formula><tex-math id="M11">\begin{document}$\delta_{u}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M11.png"/></alternatives></inline-formula> 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 (<inline-formula><tex-math id="M12">\begin{document}$u^{*}_{\theta}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M12.png"/></alternatives></inline-formula>) meets its maximum value. It is found that <inline-formula><tex-math id="M13">\begin{document}$\delta_{T}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M13.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M14">\begin{document}$\delta_{u}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M14.png"/></alternatives></inline-formula> both have a power-law dependence on the Rayleigh number. For the bubble, the scaling coefficent of <inline-formula><tex-math id="M15">\begin{document}$\delta_{T}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M15.png"/></alternatives></inline-formula> is <inline-formula><tex-math id="M16">\begin{document}$-0.32$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M16.png"/></alternatives></inline-formula> which is consistent with that from the Rayleigh-Bénard convection model. The rotation does not affect the scaling coefficent of <inline-formula><tex-math id="M17">\begin{document}$\delta_{T}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M17.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M17.png"/></alternatives></inline-formula>. On the other hand, the scaling coefficent of <inline-formula><tex-math id="M18">\begin{document}$\delta_{u}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M18.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M18.png"/></alternatives></inline-formula> equals <inline-formula><tex-math id="M19">\begin{document}$-0.20$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M19.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M19.png"/></alternatives></inline-formula> 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 <inline-formula><tex-math id="M20">\begin{document}$-0.14$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M20.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M20.png"/></alternatives></inline-formula>. The profile of <inline-formula><tex-math id="M21">\begin{document}$T^{*}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M21.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M21.png"/></alternatives></inline-formula> satisfies the scaling law of <inline-formula><tex-math id="M22">\begin{document}$T^{*}\sim\theta^{0.5}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M22.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M22.png"/></alternatives></inline-formula> with the latitude of (<inline-formula><tex-math id="M23">\begin{document}$\theta$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M23.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M23.png"/></alternatives></inline-formula>) 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 <inline-formula><tex-math id="M24">\begin{document}$0.5$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M24.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M24.png"/></alternatives></inline-formula> and shorterning the interval of satisfying the power law. Finally, it is found that the internal thermal dissipation rate and kinetic dissipation rate <inline-formula><tex-math id="M25">\begin{document}$\varepsilon^0_T$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M25.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M25.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M26">\begin{document}$\varepsilon^0_u$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M26.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M26.png"/></alternatives></inline-formula> are one order larger than their peers: the external thermal dissipation and kinetic dissipation rates <inline-formula><tex-math id="M27">\begin{document}$\varepsilon^1_T$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M27.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M27.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M28">\begin{document}$\varepsilon^1_u$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M28.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220693_M28.png"/></alternatives></inline-formula> 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.