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

Abstract

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 δ ∈ [0 • , 90 • ], the Rayleigh number is varied between Ra ∈ [3 × 10 6 , 3 × 10 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 ∝ Ra 0.3 for the DPR and Nu ∝ Ra 0.24 for the SPR. Concerning Re, the scaling in the DPR lies between Re ∝ Ra 0.48 and Re ∝ Ra 0.53 depending on Ra and δ , while in the SPR, the scaling lies between Re ∝ Ra 0.44 and Re ∝ Ra 0.45 depending on δ . The turbulent thermal and kinetic energy dissipation rates (ε T and ε u , respectively) are also very different in the DPR and SPR. The probability density functions (PDF) of the normalized log ε T and log ε u are close to a Gaussian PDF for small fluctuations, but deviate considerably from a Gaussian at large fluctuations in the DPR. In the SPR, the PDFs of normalized log ε T and log ε u deviate considerably from a Gaussian PDF even for small values. The globally averaged thermal energy dissipation rate due to the mean temperature field was shown to exhibit the scaling ε T B ∝ Ra −0.23 in the DPR, and ε T B ∝ Ra −0.28 in the SPR. The globally averaged kinetic energy dissipation rate due to the mean velocity field is shown to exhibit the scaling ε u B ∝ Ra −0.47 in the DPR (the exponent reduces from 0.47 to 0.43 as δ is increased up to 30 • ). In the SPR, the behavior changes considerably to ε u B ∝ Ra −0.27 . For the turbulent dissipation rates, the results indicate the scaling ε T B ∝ Ra −0.18 and ε u B ∝ Ra −0.29 in the DPR. However, the dependencies of ε T B and ε u B on Ra cannot be described by power-laws in the SPR. I. INTRODUCTIONTurbulent thermal convection is ubiquitous in nature and plays a significant role in large scale flows on the Earth, such as the cyclones in the atmosphere and the circulation of the deep oceans 1 . Convective flows are also vital for a great number of industrial applications, for example cooling systems on chip-boards 2 . The fluid motion in turbulent thermal convection is driven by buoyancy which arises due to temperature gradients imposed by boundary conditions 3 . In these flows, the a) Also at