A fourth-order finite difference algorithm is developed for the direct simulation of two-dimensional Rayleigh-Bénard convection in a horizontally periodic domain of aspect ratio $\Gamma$. The free-slip condition prevents the formation of kinetic boundary layers and allow the implementation of an efficient long-stencil scheme for the vorticity equation without additional risk of instability. The required grid size to properly resolve Batchelor’s microscale and therefore avoid aliasing is expressed in terms of the Rayleigh ($\mathrm{Ra}$) and Prandtl ($\mathrm{Pr}$) numbers. It was verified that the method was able to reproduce the Nusselt and Reynolds number scalings as well as the different flow regimes documented in the literature for $\Gamma =5$, $\mathrm{Pr}=10$, and $\mathrm{Ra}\le {10}^7$. Furthermore, the analytical solution of the Poisson equation in Fourier series is derived and compared with the standard fast Poisson solver. The horizontal wavenumbers decay much slower than the vertical ones, which might be explained by the adopted domain’s aspect ratio. The results also suggest that the largest energy-containing wavenumbers scale with ${\mathrm{Ra}}^{3/8}$ for large enough Rayleigh numbers.

开发了一种四阶有限差分算法，用于在纵横比的水平周期域中直接模拟二维瑞利-贝纳德对流 $\Gamma$. 自由滑移条件防止了动力学边界层的形成，并允许对涡度方程实施有效的长模板方案，而不会产生额外的不稳定风险。正确解析 Batchelor 微尺度并因此避免混叠所需的网格大小用瑞利 (Rayleigh) ($\mathrm{镭}$) 和普朗特 ($\mathrm{压力}$) 数字。经验证，该方法能够重现 Nusselt 和 Reynolds 数标度以及文献中记录的不同流态$\Gamma =5$, $\mathrm{压力}=10$， 和 $\mathrm{镭}\le {10}^7$. 此外，推导了傅里叶级数泊松方程的解析解，并与标准的快速泊松求解器进行了比较。水平波数的衰减比垂直波数慢得多，这可能是由采用的域的纵横比来解释的。结果还表明，最大的含能量波数与${\mathrm{镭}}^{3/8}$ 对于足够大的瑞利数。