傾斜槽內溶質垂直分層流體層在溫度梯度作用下之雙擴散對流穩定特性分析

Abstract

本論文建立了傾斜槽系統，其中槽高無限高，並以側向加熱的方式進行物理模型建構。其中，容器內液體存在著垂直濃度梯度，當 θ=0^° 為典型垂直槽系統，其中左壁為高溫、右壁為低溫，當角度 θ>0^° 時，下板為加熱壁；反之，當 θ<0^° 時，上板為加熱壁。本文是以垂直槽中的熱對流題目為啟發，加入傾角與模態轉換定義，並將分析結果與過往論文進行交叉比對，進而得到新的結論。 本論文旨在對自然對流系統進行線性穩定性分析。基於Boussinesq approximation的控制方程組，求解了基態解。隨後引入微小擾動，並透過正規模態展開分析其隨時間的發展，進而探討系統的穩定性。數值求解採用切比雪夫配置法計算特徵值。最終，根據所得中性穩定曲線繪製穩定性邊界圖，並與以往實驗結果對比，歸納出相應結論。 固定 Pr=6.7、Le=100 來模擬海洋情形，並修正了前人論文的錯誤。本論文引入了五個穩定模式，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)。在引入模態後，本文定義了穩定邊界圖的轉換點，並追尋了Wu and Chen所定義的模態轉換。本論文使用T2～T4 模態轉換點，並求得出漸進關係式，發現當穩定邊界圖過了最低點後，會隨著角度越大越來越穩定，也就是論文中所探討的大濃度梯度區間。也求得最低點臨界值與角度的漸進線關係式，然而本論文發現小濃度梯度與大濃度梯度呈現相反現象，因此最後探討小濃度梯度與角度變化造成的影響。本研究找到了關鍵角度 θ=〖65.8〗^° ，當角度增加到 〖65.8〗^° 時，系統會變得更加不穩定，也探討當角度為 〖65.8〗^° 時，跳動點會在 R_s=33.9 時消失。

This paper establishes a tilted slot system, in which the slot height is infinitely high, and a physical model is constructed using lateral heating. In this system, there exists a vertical concentration gradient in the liquid inside the container. When θ=0^° , it represents a typical vertical slot system, with the left wall being high temperature and the right wall being low temperature. When the angle θ>0^° , the bottom plate is the heated wall; conversely, when θ<0^° , the top plate is the heated wall. This paper is inspired by the topic of thermal convection in vertical slots, adding the definition of inclination and mode transition, and comparing the analysis results with previous papers to obtain new conclusions. The purpose of this paper is to conduct linear stability analysis of natural convection systems. Based on the control equations of the Boussinesq approximation, the base state solution is solved. Subsequently, small disturbances are introduced, and their temporal development is analyzed through a normal mode expansion to explore the stability of the system. The numerical solution uses the Chebyshev collocation method to calculate eigenvalues. Finally, based on the obtained neutral stability curve, a stability boundary diagram is drawn and compared with previous experimental results to summarize corresponding conclusions. The simulation of oceanic conditions is carried out with fixed Pr=6.7、Le=100 , and errors in previous papers are corrected. This paper introduces five stable modes, including temperature buoyancy mode (TBM), diffusion buoyancy mode (DBM), diffusion shear mode (DSM), salt finger shear mode (SSM), and salt finger buoyancy mode (SBM). After introducing the modes, this paper defines the transition points of the stable boundary diagram and explores the mode transition defined by Wu and Chen. This paper uses the T2～T4 mode transition points and obtains an asymptotic relationship, finding that as the stable boundary diagram passes its lowest point, it becomes increasingly stable with larger angles, which corresponds to the interval of large concentration gradients discussed in the paper. The paper also derives an asymptotic line relationship between the critical value at the lowest point and the angle. However, it is found that small concentration gradients and large concentration gradients exhibit opposite phenomena, so the paper finally explores the effects of changes in angle on small concentration gradients. This study identifies a critical angle of θ=〖65.8〗^° , at which the system becomes more unstable as the angle increases to 〖65.8〗^° .It also discusses that when the angle is 〖65.8〗^° , the jumping point disappears at R_s=33.9.