三維平行平板底部加熱之可壓縮自然對流現象研究

Abstract

本研究利用數值方法研究三維平行平板底部加熱之可壓縮自然對流現象研究。流場利用有限差分法進行計算，計算方法可分為兩部分：第一部份為非黏滯性項的尤拉方程式採用Roe方法計算通量，並且加入Preconditioning矩陣，讓程式在計算低速可壓縮流可獲得良好之收斂結果；第二部份為黏滯性項的計算，採用二階中央插分法。在時間項方面則採用LUSGS隱式法。出口設非反射性邊界條件避免可壓縮流中壓力波的干擾。在許多應用例子中，溫差常常大於30K，因此Boussinesq assumption不適用。 由數值計算的結果得知，流體的溫度受到加熱，體積開始膨脹，兩端的流體被擠出至大氣環境，加熱壁面附近開始產生迴流，在平行平板中，上方流體從平行平板流向大氣環境，下方則從大氣環境流向平行平板內，分佈延伸至平行平板兩端。此種邊界上的現象稱之為雙反射現象(dual-reflection phenomenon)。因邊際效應的關係，底部兩側紐塞數分佈較高，而越接近中心而紐塞數分佈逐漸降低，而隨著加熱長度與兩平板間距寬度比的增加，流場漸不穩定，但因加熱區域大，累積熱量多，故局部紐塞數小，熱傳效果越不好。

An investigation of natural convection of compressible flow in a three dimensional Parallel plate with bottom side heated . The finite difference is adopted and the computational approaches are divided into two parts. One is the inviscid terms. The Roe scheme is utilized to deal with the flux of inviscid terms and the preconditioning matrix is added to let the scheme to be more effective for all speed filed. The other is viscous term and the central difference of second order is utilized to handle it. The temporal term is solved by LUSGS. Non-reflection conditions at the outlet is derived in order to resolve reflections induced by acoustic waves. In many other important natural convection problems, the temperature differences are often higher than 30 K, Boussinesq assumption is not inapplicable. Based on the numerical results, the volume of fluid is expended due to the rising temperature, and thus leads to the fluid leaving the control volume on both ends. Then a backflow region is observed, fluid near the top half of the parallel plate flows outward, and flows inward near the bottom which is names dual-reflection phenomenon. And because of the boundary conditions, the distribution of nueeslt number of the bottom side is higher near the ends, decreases as to the center. The flow is unsteady as the heated length to the height between two parallel plate ratio gets higher. However, as the heated region gets larger, heat is accumulated, which leads to the local nusselt number decreasing and a lower heat transfer rate.