本研究對帶有隨機粗糙度元素的垂直板在瑞利數(Rayleigh number)設定為106 的自然對流條件下進行模擬，表面上的粗糙度呈現隨機分布的特徵，這可以使模擬更加符合現實中的粗糙表面，例如柏油路、樹皮等。為了處理低速可壓縮流中的自然對流問題，我們的求解器將流體視為可壓縮流體以解決完整的 Navier-Stokes 方程式，並實現高溫差的計算，使結果符合現實情況。
與過去研究認為粗糙表面可以增加熱傳的結果不同，我們的結果表明，粗糙度元素越密集的表面其平均紐森數(Nusselt number)越低，透過速度場分析，我們發現在最粗糙表面上，有更多的流體被限制在山谷裡，且山峰後面所產生的回流也相對較弱，這使得回流難以帶走山谷中的流體，降低整體散熱效果。
所有粗糙表面的局部紐森數(local Nusselt number)皆高於平板，但僅有稍微粗糙的表面，其平均紐森數高於平板。以整體速度分布結果中，我們觀察到在上游部分粗糙板的速度曲線皆低於平板，到了中下游部分，稍微粗糙的表面的速度才開始高於平板，這個現象也反映至上下游的平均紐森數分布上。在上下游中，粗糙度元素的排列對決定了該區域的熱傳好壞，而在中游處紐森數更容易受到速度快慢影響而具有更劇烈的變化。

The present study simulates vertical plates with randomly roughened elements under natural convection conditions with a Rayleigh number set at 106. The roughness on the surface exhibits characteristics of random distribution, enhancing the simulation's fidelity to real-world rough surfaces such as asphalt roads and tree bark. To address natural convection problems in low-speed compressible flows, our solver treats the fluid as compressible to solve the complete Navier-Stokes equations, enabling computations with high temperature differentials to align with real-world scenarios.
In contrast to past studies suggesting that rough surfaces enhance heat transfer, our findings indicate that surfaces with denser roughness elements exhibit lower average Nusselt numbers. Through velocity field analysis, we observed that on the roughest surfaces, more fluid is confined within valleys, and the recirculation generated behind peaks is relatively weaker. This impedes the removal of fluid from the valleys, thus reducing overall heat dissipation efficiency.
The local Nusselt numbers of all rough surfaces are higher than that of the flat plate, but only slightly rough surfaces exhibit higher average Nusselt numbers than the flat plate. In the overall velocity distribution results, we observed that the velocity curves of slightly rough surfaces in the upstream portion are lower than those of the flat plate. However, in the middle to downstream portions, the velocity of slightly rough surfaces begins to exceed that of the flat plate. This phenomenon is also reflected in the distribution of average Nusselt numbers upstream and downstream. In the upstream and downstream, the arrangement of roughness elements determines the effectiveness of heat transfer in that area, while in the middle reaches, the Nusselt number is more susceptible to changes in velocity and exhibits more pronounced variations.

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