橫向加熱驅動垂直分層流體之雙擴散對流穩定性分析

Abstract

本論文建立一套高寬比相當大的垂直槽系統，利用側向加熱的方式，達到左壁低溫、右壁為高溫，其中，容器內液體存在著垂直濃度梯度，藉以模擬冰山附近的溫鹽環流所造成的雙擴散對流之流體穩定性分析，本文係以垂直槽中的熱對流題目為啟發，再於其中加入垂直濃度梯度液體，將此分析結果與過往論文進行加叉比對，進而得到新的結論。 吾人以時域的線性穩定性分析此題目，從Boussinesq approximation的統御方程式起手，求解出基態解，再以此基態解加入微小擾動，並以正規模態展開的方式分析此微小擾動隨著時間的發展情形，藉此分析此系統之穩定性狀態；數值方法則是利用切比雪夫配置法來求解系統特徵值，最後，以蒐集出來的中性曲線圖，繪製出穩定性邊界圖，將此圖結果與其他論文比較，歸納出結論。 結果顯示，在Pr=7、Le=100的案例中，吾人所作之結果相較於本研究所比較的論文有更高的準確性，並透過蒐集到相當多且有利的數據，將系統的穩定模式區分為五個階段，包含:溫度浮力模式(TBM)、擴散浮力模式(DBM)、擴散剪力模式(DSM)、鹽指剪力模式(SSM)以及鹽指浮力模式(SBM)等。之後，分析不同普朗特數和路易斯數會如何影響系統的穩定模式，再以上述五種不同模式為基礎，將普朗特數影響區分為三個區塊，並同樣發現再普朗特數為12.5時，系統會發生模態轉換，這與過去論文所得結論相同；至於路易斯數影響部分，吾人同樣將其區分為兩個區塊，於Le=1 & 2 區塊中，找尋到新的漸近線關係式，在 Le>2 區塊中，則是驗證了過去所得的漸近線關係式之正確性。

A system of a vertically oriented slot with a large aspect ratio is considered in this study. The slot is subjected to lateral heating on both sides, resulting low temperature in left-hand side compared to the right-hand side, which is maintained at a higher temperature. Additionally, a constant vertical salinity gradient is imposed on the liquid inside the slot. The objective of this research is to analyze the stability of double-diffusive convection near the iceberg. This study was motivated by the problem of heat convection in a vertical slot and we extends it by incorporating a vertical salinity gradient. The obtained results are compared with those presented in previous studies, leading to novel conclusions. In this study, we employed the concept of temporal instability to analyze the problem. Initially, the Boussinesq approximation was applied to the governing equation, which allowed us to determine the basic state solution. Subsequently, the linearized perturbation equation was utilized to investigate the instability of the system. To facilitate the analysis, we expanded the perturbation equation using the normal mode expansion. For numerical calculation, the Chebyshev collocation method was employed to compute the eigenvalues of the equation. By generating extensive neutral curves, we were able to plot the stability boundary and compare our findings with those reported in other references. Through this comparative analysis, we derived novel and insightful conclusions. Our results revealed the presence of five distinct stability modes in the case of a Prandtl number of 7 and a Lewis number of 100. This finding was made possible by utilizing a more precise approach and analyzing a vast amount of data. Additionally, we examined the influence of different Prandtl and Lewis numbers on the system's stability. Based on the five stability modes mentioned earlier, we identified three regions that correspond to the effect of the Prandtl number on the stability boundary. Notably, we observed a mode transition at a Prandtl number of 12.5, which aligns with the findings of previous studies. Regarding the effect of the Lewis number, we distinguished it into two parts. In the region where the Lewis number is equal to 1 and 2, we derived a new asymptotic solution. In the region where the Lewis number exceeds 2, we validated the results of the asymptotic solution proposed in previous papers. These findings contribute to a deeper understanding of the system's stability behavior under varying Prandtl and Lewis numbers.