The focus of this research is to delineate the thermal behavior of a rarefied monatomic gas confined between horizontal hot and cold walls, physically known as rarefied Rayleigh-Bénard (RB) convection. Convection in a rarefied gas appears only for high temperature differences between the horizontal boundaries, where nonlinear distributions of temperature and density make it different from the classical RB problem. Numerical simulations adopting the direct simulation Monte Carlo approach are performed to study the rarefied RB problem for a cold to hot wall temperature ratio equal to $r=0.1$ and different rarefaction conditions. Rarefaction is quantified by the Knudsen number, Kn. To investigate the long-time thermal behavior of the system two ways are followed to measure the heat transfer: (i) measurements of macroscopic hydrodynamic variables in the bulk of the flow and (ii) measurements at the microscopic scale based on the molecular evaluation of the energy exchange between the isothermal wall and the fluid. The measurements based on the bulk and molecular scales agreed well. Hence, both approaches are considered in evaluations of the heat transfer in terms of the Nusselt number, Nu. To characterize the flow properly, a modified Rayleigh number (${\mathrm{Ra}}_m$) is defined to take into account the nonlinear temperature and density distributions at the pure conduction state. Then the limits of instability, indicating the transition of the conduction state into a convection state, at the low and large Froude asymptotes are determined based on ${\mathrm{Ra}}_m$. At the large Froude asymptote, simulations following the onset of convection showed a relatively small range for the critical Rayleigh (${\mathrm{Ra}}_m=1770\pm 15$) that flow instability occurs at each investigated rarefaction degree. Moreover, we measured the maximum Nusselt values ${\mathrm{Nu}}_{\max }$ at each investigated Kn. It was observed that for $\mathrm{Kn}\ge 0.02, {\mathrm{Nu}}_{\max }$ decreases linearly until the transition to conduction at $\mathrm{Kn}\approx 0.03$, known as the rarefaction limit for $r=0.1$, occurs. At the low Froude (parametric) asymptote, the emergence of a highly stratified flow is the prime suspect of the transition to conduction. The critical ${\mathrm{Ra}}_m$ in which this transition occurs is then determined at each Kn. The comparison of this critical Rayleigh versus Kn also shows a linear decrease from ${\mathrm{Ra}}_m\approx 7400$ at $\mathrm{Kn}=0.02$ to ${\mathrm{Ra}}_m\approx 1770$ at $\mathrm{Kn}\approx 0.03$.

中文翻译：

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稀薄气体条件下瑞利-贝纳德对流传热的数值研究。

这项研究的重点是描绘限制在水平热壁和冷壁之间的稀有单原子气体的热行为，物理上称为稀有瑞利-贝纳德（RB）对流。稀有气体中的对流仅在水平边界之间存在高温差异时出现，其中温度和密度的非线性分布使其不同于传统的RB问题。进行了采用直接模拟蒙特卡洛方法的数值模拟，以研究冷热壁温度比等于的稀疏RB问题。$\mathrm{[R}=0.1$和不同的稀疏条件 通过克努森数Kn定量反射。为了研究系统的长期热行为，遵循以下两种方法来测量传热：（i）测量大部分流量中的宏观流体力学变量，以及（ii）基于对分子的分子评估在微观尺度上的测量等温壁与流体之间的能量交换。基于体积和分子规模的测量结果非常吻合。因此，两种方法都可以根据努塞尔特数Nu进行传热评估。为了正确表征流量，请修改瑞利数（${镭}_{米}$定义）是为了考虑纯传导状态下的非线性温度和密度分布。然后，根据以下公式确定在低弗鲁德渐近线和大弗鲁德渐近线处的不稳定性极限（表明传导状态已转变为对流状态）${镭}_{米}$。在大Froude渐近线处，对流开始后的模拟显示临界瑞利（${镭}_{米}=\mathrm{1770年}\pm 15$）在每个研究的稀疏度下都会发生流动不稳定性。此外，我们测量了最大的Nusselt值${\mathrm{怒江}}_{\mathrm{最高}}$在每个调查的Kn。据观察，对于$n\ge 0.02， {\mathrm{怒江}}_{\mathrm{最高}}$ 线性减小，直到在 $n\approx 0.03$，称为 $\mathrm{[R}=0.1$，发生。在低弗洛德（参数）渐近线处，高度分层流动的出现是向传导过渡的主要怀疑。关键${镭}_{米}$然后在每个Kn处确定发生这种转变的位置。该临界瑞利与Kn的比较还显示出与${镭}_{米}\approx 7400$ 在 $n=0.02$ 至 ${镭}_{米}\approx \mathrm{1770年}$ 在 $n\approx 0.03$。