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.

Traditional methods are difficult to deal with solid propellant regression in complex geometry, especially for a finocyl grain with cracks. In this work, the Level-Set method is adapted to solid propellant regression. The advantage of this method is the implicit representation of interface and no restriction on geometrical configuration. Therefore, tiny cracks can be easily captured by the Level-Set method without any complex grids. A three-dimensional solid propellant regression evolution process of a finocyl grain is presented. A one-dimensional unsteady internal ballistics numerical method is applied to obtain pressure and thrust time histories, coupled with the three-dimensional grain burnback model. The mass and energy injection in the governing equations can be computed from the Level-Set formulation. And the propellant burning rate can be computed from the local pressure. The results show that a crack increases the head-end pressure and thrust at early stage, because of the increment of the burning surface area. It can be seen from the evolution of propellant regression that the crack gradually expands.

No abstract

We use 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∈[0o,90o], the Rayleigh number is varied between Ra∈[3*106, 3*109]. 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 and Reynolds number are different in these two regimes, with Nu∝Ra0.3 for the DPR and Nu∝Ra0.24 for the SPR. Concerning Re, the scaling in the DPR lies between Re∝Ra0.48 and Re∝Ra0.53 depending on Ra and δ, while in the SPR, the scaling lies between Re∝Ra0.44 and Re∝Ra0.45 depending on δ. The turbulent thermal and kinetic energy dissipation rates (εT' and εu', respectively) are also very different in the DPR and SPR. The PDFs of the normalized logεT' and logεu' 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 logεT' and logεu' 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 different scaling.