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

Abstract

The soap bubble heated at the buttom is a novel thermal convection cell, which has the inherent spherical surface and quasi two-dimensional features, so that it could provide insights 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 and viscous boundary layers, the thermal and kinetic dissipation by Direct Numerical Simulation (DNS). The thermal and kinetic dissipations are mostly occured in the boundary layers, it reveals the most significance of the boundary layers during the process of the energy absorption. Considering the complex characteristics of the heated bubble, this study proposes the new definition to identify the thermal and viscous boundary layer. The thermal boundary layer thicknesse $\delta_{T}$ is defined as the geodetic distance between the equator of the bubble and the latitude at which the the r.m.s temperature ($T^{*}$) reaches the maximum. On the other hand, the viscous boundary layer thicknesse $\delta_{u}$ is the geodetic distance from the equator at the latitude where the extrapolation for the linear part of the r.m.s turbulent latitude velocity ($u^{*}_{\theta}$) meets its maximum. It is discovered that $\delta_{T}$ and $\delta_{u}$ 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 the Rayleigh-Bénard convection The rotation do not effect the scaling coefficent of $\delta_{T}$. On the other hand, the scaling coefficent of $\delta_{u}$ equals to $-0.20$ and is different from the Rayleigh-Bénard convection. The weak rotation do not change the coefficent while the strong rotation increase it to $-0.14$. The profile of $T^{*}$ satisfies the scaling law of $T^{*}\sim\theta^{0.5}$ with the latitude ($\theta$) on the bubble. The scaling law of the r.m.s temperature profile coincides with the theoretical prediction and the results obtained from the Rayleigh-Bénard convection. 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. At last, it is discovered that the internal thermal and kinetic dissipation rates $\varepsilon^0_T$ and $\varepsilon^0_u$ are one order larger than their peers:the external thermal and kinetic dissipation rates $\varepsilon^1_T$ and $\varepsilon^1_u$ based on a thorough analysis of the energy budget. The major thermal and kinetic dissipation are accumulated in the boundary layers. With the increasing rotation rate, fewer energy is transfered from the buttom to the top of the bubble and the influence of the external energy dissipations is less pronouced.