Rough-surface Rayleigh-Bénard convection is investigated using direct numerical simulations in two-dimensional convection cells with aspect ratio Γ=2. Three types of fractal roughness elements, which are marked as n1, n2 and n3, are constructed based on the Koch curve and sparsely mounted on both the plates, where n denotes the level of the roughness. The considered Rayleigh numbers Ra range from 10⁷ to 10¹¹ with Prandtl number Pr =1. Two regimes are identified for cases n1, n2. In Regime I, the scaling exponents β in the effective Nusselt number Nu vs Ra scaling Nu ∼ Ra^β reach up to about 0.4. However, when Ra is larger than a critical value Ra_c, the flow enters Regime II, with β saturating back to a value close to the smooth-wall case (0.3). Ra_c is found to increase with increasing n, and for case n3, only Regime I is identified in the studied Ra range. The extension of Regime I in case n3 is due to the fact that at high Ra, the smallest roughness elements can play a role to disrupt the thermal boundary layers. The thermal dissipation rate is studied and it is found that the increased β in Regime I is related with enhanced thermal dissipation rate in the bulk. An interesting finding is that no clear convection roll structures can be identified for the rough cases, which is different from the smooth case where well-organized convection rolls can be found. This difference is further quantified by the detailed analysis of the plume statistics, and it is found that the horizontal profiles of plume density and velocity are relatively flattened due to the absence of clear convection rolls.

在纵横比Γ = 2的二维对流单元中使用直接数值模拟来研究粗糙表面Rayleigh-Bénard 对流。三种类型的分形粗糙度单元，分别标记为 n1、n2 和 n3，它们是基于Koch曲线构建的，并且稀疏地安装在两个板上，其中 n 表示粗糙度的水平。所考虑的瑞利数Ra范围从 10 ⁷到 10 ^{11 ，}其中普朗特数Pr =1。为案例 n1、n2 确定了两种制度。在规则 I 中，有效 Nusselt 数Nu 中的标度指数β与Ra标度Nu ∼Ra ^β达到约0.4。然而，当Ra大于临界值Ra _{c 时}，流动进入状态 II，β饱和回到接近光滑壁情况 (0.3) 的值。发现Ra _c随着 n 的增加而增加，并且对于情况 n3，在研究的Ra范围内仅确定了制度 I。在 n3 的情况下，Regime I 的扩展是由于在高Ra 下，最小的粗糙度元素可以起到破坏热边界层的作用。研究了热耗散率，发现增加的β状态 I 与增强的散装热耗散率有关。一个有趣的发现是在粗糙的情况下无法识别出清晰的对流卷结构，这与可以找到组织良好的对流卷的平滑情况不同。通过对羽流统计的详细分析进一步量化了这种差异，发现由于没有明显的对流卷，羽流密度和速度的水平剖面相对平坦。