非軸對稱系統對環流雙擴散穩定性的影響

Abstract

本研究旨在探討垂直高同心圓環間隙中給予徑向溫度梯度和濃度梯度的系統的穩定性。我們通過三維線性穩定性分析，對軸對稱系統和非軸對稱系統進行了詳細研究。研究結果顯示，系統的穩定性可歸納為溫度擴散區熱對流型、鹽指區鹽指型和雙擴散區雙擴散型三種不同類型。本研究經由穩定性計算與推導和Matlab數值分析，收集數據並整理分析出與之相關的討論。 本研究藉由時間的線性穩定性理論，探討在軸對稱模式與非軸對稱模式中，改變方位角波數(𝑚)、內外圓柱的半徑比(𝜂)、普朗特數(𝑃𝑟)、路易斯數(𝐿𝑒)，與溫度差和濃度差之間對系統造成的不穩定性影響。透過改變方位角波數(𝑚)、內外圓柱的半徑比(𝜂)，了解穩定性邊界產生的變化。 透過固定熱格拉曉夫數(𝐺𝑟)和溶質格拉曉夫數(𝐺𝑠)，來研究雙擴散區以及溫度擴散區，對於軸對稱模式與非軸對稱模式的穩定性變化。以及改變普朗特數(𝑃𝑟)對系統整體的不穩定性影響，和改變路易斯數(𝐿)對於鹽指區和雙擴散區的不穩定性變化。通過計算，了解系統的穩定性對於鹽指區、溫度擴散區和雙擴散區的變化與影響。 從研究結果顯示，非軸對稱系統在𝜃方向波數增加和半徑比減少的情況下，系統的穩定性會在此時增加。軸對稱系統中，減小𝜂或是加大圓環間隙寬度會使系統的穩定性減少。並且不論是改變𝜂、𝑃𝑟和𝐿𝑒，非軸對稱系統總是會比軸對稱系統來得更加穩定。

This study aims to investigate the stability of a system with radial temperature and concentration gradients within vertically aligned concentric annular gaps. We conducted a comprehensive analysis using three-dimensional linear stability analysis on both axisymmetric and non-axisymmetric systems. Research results show that the stability of the system can be summarized into three different types: thermal convection type in the temperature diffusion region, salt finger type in the salt finger region, and double diffusion type in the double diffusion region. Through stability computations, derivations, and numerical analysis using Matlab, we collected data and systematically analyzed and discussed the relevant findings. This study uses the linear stability theory of time to explore how to change the azimuth wave number (m) the radius ratio of the inner and outer cylinders (η) the Prandtl number (Pr) and the Lewis number (Le) in the axisymmetric mode and the non-axisymmetric mode and the instability effect on the system caused by temperature differences and concentration differences. By changing the azimuth wave number (m) and the radius ratio of the inner and outer cylinders (η) we can understand the changes in the stability boundary. By fixing the thermal Grashof number (Gr) and the solute Grashof number (Gs), the stability changes of the double diffusion region and the temperature diffusion region for the axisymmetric mode and the non-axisymmetric mode are studied. As well as the instability effect of changing the Prandtl number (Pr) on the overall system, and the instability changes of the salt finger region and double diffusion region by changing the Lewis number (Le). Through calculation, understand the changes and effects of system stability on the salt finger region, temperature diffusion region and double diffusion region. The results of the study indicate that in non-axisymmetric systems, stability increases with the increase in wavenumber in the θ-direction and the decrease in radius ratio. In axisymmetric systems, reducing η or increasing the annular gap width decreases the system's stability. Furthermore, regardless of changes in η, Pr, and Le, non-axisymmetric systems are always more stable than axisymmetric systems.