Numerical Simulation of the Quasi-2D Turbulence on a Half Soap Bubble Heated at the Bottom

Abstract

The soap bubble heated at the bottom is introduced as a novel quais two-dimensional turbulence. The curved geometry of the bubble brings the challenges for the Direct Numercial Simulations(DNS) of the turbulence on the bubble. In oder to overcome the difficulties due to the curved geometry, a numerical method based on the stereographic projection is implemented for the DNS of the soap bubble. The numerical method to compute the spectrum, the flux and the structure functions of the flows on the bubble are introduced in detail. Three different Rayleigh numbers at <inline-formula><tex-math id="M3">\begin{document}$Ra=3\times10^{7}, 3\times10^{9}, 3\times10^{11}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="amm-43-0143_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="amm-43-0143_M3.png"/></alternatives></inline-formula> are simulated based on present numerical methods. And the related spectrum, flux and structure functions are calculated. It is indicated that both the inversed energy cascade and forward enstrophy cascade are observed from all of the calculated cases. <inline-formula><tex-math id="M4">\begin{document}${\rm{Bo}}59$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="amm-43-0143_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="amm-43-0143_M4.png"/></alternatives></inline-formula> scaling law is found to fit the small scale fluctuations on the soap bubble. As the increase of Rayleigh number, the turbulent energy reduces for the large scale plumes, and the kinetic energy increases for the higher wave number structures.