The effect of tilt on turbulent thermal convection for a heated soap bubble

We use direct numerical simulation (DNS) to explore the effect of tilt on two-dimensional turbulent thermal convection on a half-soap bubble that is heated at its equator.In the DNS, the bubble is tilted by an angle $\delta\in[0^{\circ},90^{\circ}]$, the Rayleigh number is varied between $Ra\in[3\times10^6, 3\times10^9]$, and the Prandlt number is fixed at $Pr=7$.The DNS reveals two qualitatively different flow regimes: the dynamic plume regime (DPR) and the stable plume regime (SPR).In the DPR, small dynamic plumes constantly emerge from random locations on the equator and dissipate on the bubble.In the SPR, the flow is dominated by a single large and stable plume rising from the lower edge of the bubble.The scaling behaviour of the Nusselt number $Nu$ and Reynolds number $Re$ are different in these two regimes,with $Nu\propto Ra^{0.3}$ for the DPR and $Nu\propto Ra^{0.24}$ for the SPR. Concerning $Re$, the scaling in the DPR lies between $Re\propto Ra^{0.48}$ and $Re\propto Ra^{0.53}$ depending on $Ra$ and $\delta$,while in the SPR, the scaling lies between $Re\propto Ra^{0.44}$ and $Re\propto Ra^{0.45}$ depending on $\delta$.The turbulent thermal and kinetic energy dissipation rates ($\epsilon_{T^{\prime}}$ and $\epsilon_{u^{\prime}}$, respectively) are also very different in the DPR and SPR.The probability density functions (PDF) of the normalized $\log\epsilon_{T^{\prime}}$ and $\log\epsilon_{u^{\prime}}$ are close to a Gaussian PDF for small fluctuations, but deviate considerably from a Gaussian at large fluctuations in the DPR.