本論文以研究二、三維方形空穴內自然對流的行為，經由三維數值的計算與層紊流之判斷方法，區分出空穴中層紊流的區域，從三維的數值平台模擬臨界瑞里數，建立紊流分析所需的資料庫，進而評估紊流區域內之紊流特質。本論文以空氣為介質，研究瑞里數於105~6×108幾種流況。數值方法則以步進(fractional step method)演算法與不加任何假設的統御方程式(exact formulation)，求得壓力的Poisson方程式，並配合理想氣體方程式獲得密度與速度等值，在時間項與非線性對流項則分別以二階準確Adams -Bashforth方法與四階插分方法處理之。
本論文可以分成以下幾項目標，首先，發展三維非穩態之數值計算平台，驗證數學模式中使用Boussinesq假設的正確性，採用直接數值方法模擬三維方形空穴之自然對流現象，藉由三維的模擬，了解Boussinesq假設對自然對流模擬所產生的影響。第二階段發展三種判斷層、紊流之準則，使用前階段建立的三維非穩態數值計算平台，所得到的數值資料庫，判斷出流場中混沌的開始，作為使用紊流模式之參考依據。第三階段針對Ra 值為 4×108與6×108兩種流場，使用三種判斷準則，區分中間平面層、紊流發生之區域。 最後則對三維流場Ra值為6×108在紊流區內，使用前階段之數值資料進行統計分析，評估紊流強度、紊流動能與其消散率的多寡，作為了解紊流結構的參考依據。
本論文研究結果綜整如下：一、在溫差變化較小如Ra值為105的流況下，仍可使用Boussinesq假設，在溫差變化較大者如Ra值為2×108，將不再適用Boussinesq假設，以符合實際溫度差所影響的差異。二、在速度較低的流場中，流體低於Ra 值 4*108時仍可以使用二維方程式模擬流場狀態，當Ra 值 大於4*108時流體進入弱紊流，二維與三維之偏差增加，此時須要使用三維方程式來模擬，才能顯現出第三維的變化。三、依據三種層、紊流判斷準則顯示，流體在二維流場時從穩定的層流(108)，進入不穩定型態的週期流(2×108)，當Ra值為4×108時，在左上角與右下角之區域，流體已經開始呈現局部混沌行為，因此紊流區域分佈在左上角與右下角。在三維流場Ra值為6×108，紊流區域仍是分佈在左上角與右下角。四、根據三維流場Ra值為6×108之資料進行紊流強度與紊流動能的分析，在方形中間平面的區域中，整個紊流區域仍是集中在左上角與右下角的區域，這與層、紊流判斷準則所獲得的結果相同。

A numerical study on the thermally induced flow in a square cavity heated by a vertical side was conducted to identify the transitions from laminar to turbulent regimes that occur with the increase of Rayleigh numbers and to understand their underlying physical mechanism. The numerical simulation of natural convection in the literature has usually been performed using the Boussinesq approximation, while few studies have been performed employing the less restrictive low-Mach-number approximation. It would therefore be interesting to examine whether the usual approximations made in formulating the natural convection phenomena, particularly under the condition near the critical Rayleigh number (Ra), are appropriate.
The objectives of this study are fourfold. The first one is to perform numerical computations of a three-dimensional natural convection in a square cavity. The second one is to study how the flow evolves from a laminar regime, through a transition phase, to a turbulent regime as a result of increasing Ra values. The third one is to identify the distribution of laminar/turbulent flow regions for cases in which the Ra values are slightly larger than Racr. The last one is to evaluate the turbulent statistics such as mean flow properties, Reynolds stresses, and the turbulence kinetic energy and its dissipation rate, using the benchmark data obtained in the direct numerical simulation.
Model formulation is made without introducing the usual approximations. The three- dimensional, time-dependent governing equations are solved using the fractional step method. The time-advancement sequence is treated using the second-order Adams-Bashforth scheme, while the spatial discretization is made using the fourth order scheme. Three methods of analysis, the power spectrum, the phase trajectory and the largest Lyapunov exponent, are adopted to distinguish flow pattern inside the cavity.
It was found that cases with a Ra of less than 108 were steady, laminar. When the Ra reached the values of 2×108 and 3×108, the motion of the flows become periodic/period doubling motion. As Ra was increased to 4×108 or larger, the flows exhibited a chaotic motion (weak turbulence). The study showed turbulence starting to appear in the upper-left and lower-right corners of the cavity at Ra values of 4×108 and 6×108. The contours of the turbulent regions at Ra values of 4×108 and 6×108 were thus determined.
It was found that the need for proper model formulation, such as exact formulation against the usual one with the Boussinesq approximation and 3-D against 2-D formulation in the theoretical study of the present problem becomes more essential as the Ra is increased to its critical value.
Turbulence statistics including the mean flow properties, Reynolds stresses, k, , and mean vorticity, were calculated in the case of an Ra values of 6×108. It was found that the turbulent regions observed on the basis of the DNS data basically coincident with those are determined by the analytical methods of the power spectrum, the phase trajectory and the largest Lyapunov exponent.

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