圓柱環流場之雙對流穩定性分析

Abstract

本篇論文建立一套空心高圓柱體且側向有溫度及濃度梯度存在之系統，利用擁有濃度差異之流體與熱對流耦合之關係而產生的雙擴散對流效應，以往雙擴散自然對流的討論不論是實驗還是數值模擬都局限於直角座標系，而在圓柱座標系的討論中通常會加上內壁旋轉的條件使之產生泰勒渦旋(Taylor vortex)，通常實驗以及模擬結果都會強烈的受到內壁轉速影響，這個結果並不是我們所期望的。本論文希望討論雙擴散所造成的自然對流現象，然而目前模擬以及實驗僅有探討過圓柱體側向加熱並且產生自然對流，非常少數的實驗有進行雙擴散自然對流處於圓柱座標系之下的討論。本論文透過穩定性理論以及MATLAB數值分析，收集數值並且整理分析歸納出一系列的比較討論。 本研究旨在運用時間的線性穩定性理論，並探討內外圓柱半徑比例即是狹縫寬度大小、溫度差和濃度差以及軸對稱和非軸對稱對此系統的不穩定性影響。整個系統固定溶液為鹽水，取Le = 100，Pr = 7，可以發現軸對稱圓柱座標系會產生和直角坐標系穩定性邊界圖一樣的四大區域，但是其鹽指(salt-finger)區域並不明顯，並且當溫度差與濃度差相近時流體流速非常緩慢。接著比較小間距下的軸對稱與非軸對稱圓柱，可以發現非軸對稱圓柱下產生明顯的鹽指區域，並且在考慮θ方向的波數時可以觀察出，不論波數大小對整體穩定性邊界都幾乎不產生影響，且軸對稱與非軸對稱座標系溫度擴散區域會幾乎同時進入不穩定，但是對於濃度擴散部份非軸對稱較晚進入不穩定，代表濃度對軸對稱圓柱坐標系的影響比非軸對稱坐標系要來的劇烈。對於各不同半徑比可以發現半徑比越小即狹縫空間越大，非軸對稱不同波數的模型下θ方向的波數穩定性邊界會明顯地向右移動。

This paper establishes a system of tall annular with the temperature and concentration gradients are in the radial direction. The system uses the double-diffusion convection effect by coupling the fluids with different concentration and thermal to generate a convection. Previous discussion about the double-diffusion natural convection phenomenon, whether it is experimental or numerical simulation is limited to the rectangular coordinate system. However when the discussion of the cylindrical coordinate system, they usually add the condition of the inner wall rotation which will effect to produce a Taylor vortex. Experiments and simulation results are strongly affected by the internal wall rotate speed, which is not what we expected. This paper hopes to discuss the phenomenon of natural convection caused by double diffusion. However, the current references only discuss the lateral heating of the cylinder and generate natural convection. Very few experiments have discussed the double diffusion natural convection under the cylindrical coordinate system. In this paper, we use the stability theory and MATLAB numerical analysis, the numerical values are collected and analyzed to summarize a series of comparative discussions. The purpose of this study is to apply the theory of stability and to explore the influence of the ratio, the effect of temperature difference and concentration difference, and also the axisymmetric and non-axisymmetric systems. First we consider the whole system is saline, take Le = 100, Pr = 7, and compare the small space axisymmetric cylindrical coordinate system and rectangular coordinate system in transverse mode. It can be found that the temperature diffusion area of the cylindrical coordinate system is wider and the salt-finger area does not occur, and the fluid flow rate is very slow when the temperature difference is close to the concentration difference. When compare with the axisymmetric and non-axisymmetric cylinders can be found that there is a significant salt-finger region under the non-axisymmetric cylinders, and it can be observed when considering the wave number in the θ direction, regardless of the wave number size versus temperature concentration map there is no effect about non-axisymmetric system. The temperature diffusion region of the axisymmetric and non-axisymmetric coordinate systems will enter the instability at the same time, but the concentration diffusion part of the non-axisymmetric enters the instability later. For different radius ratios, it can be found that the smaller the radius ratio which also means the larger the slit space, and the difference between the concentration and temperature maps drawn by the wavenumber in the θ direction under the model of non-axisymmetric wavenumbers will become more and more obvious.