普朗特數和路易斯數對環流中雙擴散穩定性之影響

Abstract

本論文的理論模型建立在一套高同心圓柱間隙中且水平徑向具有溫度及濃度梯度存在之系統，其內圓柱維持溫度、濃度高；外圓柱維持溫度、濃度低，通過徑向溫度梯度和濃度梯度之間的相互競爭作用將會形成連續的渦流且系統伴隨著產生三種不同類型的雙擴散不穩定性，分別為溫度、鹽指及雙擴散。本文藉由穩定性理論推導和Matlab數值分析方法，收集數據並整理分析出一套此系統的參數圖。 本研究利用時間不穩定性(temporal instability)的線性穩定性理論，探討在內外圓柱軸對稱的情況下，改變普朗特數(Pr)、路易斯數(Le)、內外圓柱半徑比以及溫度、濃度差對於此系統的不穩定性影響。透過固定Le改變Pr以及固定Pr改變Le且固定不同的半徑比進行計算，以龐大的數據量支持，可以發現所得到的穩定性邊界圖會分成三大區域，分別為溫度擴散區(thermal diffusive)、鹽指區(salt-finger)、雙擴散區(double-diffusive)。 透過溫度擴散區以及雙擴散區的數據發現此系統存在模態的轉換，系統會隨著Pr以及Le的增加，從臨界波數較大的剪力模態(shear mode)轉換為臨界波數較小的浮力模態(buoyancy mode)且中性曲線的極值發生跳動的改變，故本研究著重於分析此二區域，在固定Le改變Pr的情況下得出了Pr跳動點對於半徑比的關係式；在固定Pr改變Le的情況下得出了Le跳動點對於半徑比的關係式。

The theoretical model of this thesis is based on a system of high concentric cylindrical gaps with radial temperature and concentration gradients. The inner cylinder maintains the high temperature and the high concentration; the outer cylinder maintains the low temperature and the low concentration. The continuous vortices will be formed through the competition between the radial temperature gradient and the concentration gradient, and the system will be accompanied by three different types of double-diffusive instabilities: thermal, salt-finger and double-diffusive. In this thesis, the stability theory derivation and Matlab numerical methods are used to collect data and set up a parameter diagrams of the system. In this thesis, we using the linear stability theory of temporal instability. In the case of axisymmetric mode, the effect of changing Prandtl number(Pr), Lewis number(Le), ratio of inner and outer cylinder radii, temperature and concentration on the instability of the system is discussed. By changing Pr, fixing Le and changing Le, fixing Pr and fixing different radius ratios for calculation. With the huge amount of data support, it can be found that the stability boundary diagram will be divided into three major regions: thermal diffusive region, salt-finger region and double-diffusive region. Through the data of the thermal diffusion region and the double-diffusive region, we found that there is a transition of the mode in the system. With the increase of Pr and Le, the system will transform from the shear mode with a high critical wavenumber to the buoyancy mode with a low critical wavenumber and the extreme value of the neutral curve will jump. Therefore, this thesis focuses on the analysis of these two regions. In the case of fixing Le and changing Pr, the relational expression of Pr jumping point to radius ration is obtained; in the case of fixing Pr and changing Le, the relational expression of Le jumping point to radius ration is obtained.