臺灣博碩士論文加值系統

本研究利用廣義有限差分法(generalized finite difference method, GFDM)配合投影法(projection method)來求解多維度之自然熱對流(Natural convections)問題，而本研究所使用的數學方程式為奈維爾－史托克斯方程式(Navier-Stokes equations)，此方程式用於描述空氣或液體等流體的運動行為，本研究就是以原始變數法型態的奈維爾－史托克斯方程式來模擬多維度的自然熱對流問題。自然熱對流通常是指大氣中的對流、海洋中的洋流及地函中的熱對流，但是在日常生活中也是隨處可見，如燒開水時水壺裡面的水體流動，又或是平日使用的電腦機殼裡之空氣流動等，除了這些常見的現象之外，更可以應用到工程方面，如溫室中的氣溫調控或是精密機械的散熱設計等。本研究對於此數學方程式是採用廣義有限差分法及投影法來離散其中的空間與時間之偏微分項，進而獲得流場動態結果。
廣義有限差分法是屬於區域型的無網格法(meshless methods)，此種方法有不需要建置網格之優勢，而且因其區域型的特性，可以避免在數值求解中產生滿矩陣和病態矩陣，故可以將此方法運用在大尺度的物理問題上，也能提高電腦模擬的計算效率；而投影法可以將奈維爾－史托克斯方程式中的速度項及壓力項分離來分別求解，此種方法將原本相當困難求解的耦合方程式分成三步驟來分別求解，不僅可以增加計算效率，也可以避免求解太過複雜的計算流程，而本研究就是採用此兩種數值模擬方法，搭配Matlab的程式撰寫來建立此一無網格法數值模擬模型。
本研究的例題分為兩大部分，分別是模擬二維及三維的自然熱對流問題，首先在二維的部分，採用計算域幾何形狀不同的三個例題，其中兩個例題的結果是與前人文獻的結果做驗證，以確保模式的準確性，而第三個例題則是本研究自行設計的案例，可用以驗證模式的一致性與穩定性，而每個例題中又根據不同的四種流況做分析以驗證此模式的實用性。而驗證完二維自然熱對流的案例後，接著將模式推廣到三維的自然熱對流案例中，一樣是以計算域的形狀分成三個例題，兩個例題是與前人文獻之結果相互比對以證明模式的準確性，第三個小題為本研究自行設計的例題，藉以了解在不同加熱條件下的流場流況，也利用不同的參數來驗證此模式的穩定性。

In this thesis, we used the generalized finite difference method (GFDM) and the projection method to accurately and efficiently analyze multi-dimensional natural convection problems. The Navier-Stokes equations are adopted as the governing equation since the Navier-Stokes equations are well-known to describe the fluid dynamics such as air and liquid. Thus, we adopted the primitive-variables formulation of the Navier-Stokes equations as the governing equation to simulate multi-dimensional natural convection problems. Natural convection appeared in our daily life, such as the convection in the atmosphere, ocean current and mantle convection, even if you boil the water or use the computer will cause natural convection. It’s very important to accurately simulate natural convection problems, so we can understand some physics of natural convection, and apply it to various engineering problems such as greenhouse and thermal design. In this study, we used the GFDM and the projection method to analyze the partial differential equations of natural convection.
The GFDM is a localized domain-type meshless method, which can avoid the time-consuming task of mesh generation. The GFDM can yield sparse matrix rather than full matrix and ill-conditioning matrix. On the basis of these advantages, the GFDM can be adopted to solve problems accurately and efficiently, especially for large-scale problems. On the other hand, we can separate the velocity and pressure fields of the Navier-Stokes equations into three steps by using the projection method. The projection method can enhance the efficiency and avoid the complex calculations of coupled equations. Thus, we adopted these two methods and Matlab programming to build the meshless numerical model and to analyze multi-dimensional problems of natural convection.
In this thesis, we analyzed two-dimensional and three-dimensional natural convection problems. The accuracy and efficiency of the numerical model can be verified by numerical comparisons in these examples. For the two-dimensional natural convection problems, we provided three examples. The first two examples are used to verify that this proposed numerical model can accurately solve the problem by comparing with results in the past study. The computational domain of the third example is designed by our own to test the stability and consistency of this numerical model. We also adopted different Rayleigh numbers in each example to validate that the proposed numerical model is suitable for various kinds of flow fields. After the numerical simulation of two-dimensional problems, we extended the proposed numerical model to accurately analyze three-dimensional problems. By testing different parameters in the proposed meshless numerical model, we can assure that this proposed meshless numerical model in this thesis is very stable and accurate as well as has great potential to be extended to realistic engineering problems of natural convection.

摘要 I
Abstract II
目錄 IV
圖目錄 V
第一章 導論 1
1.1研究目的 1
1.2 文獻回顧 2
1.2.1奈維爾－史托克斯方程式 2
1.2.2自然熱對流 3
1.2.3投影法 4
1.2.4無網格法 5
1.2.5廣義有限差分法 8
第二章 物理問題與控制方程式 10
2.1奈維爾－史托克斯方程式 10
2.2控制方程式無因次化 16
2.3自然熱對流 17
2.4邊界條件 18
第三章 數值方法 20
3.1投影法 20
3.2廣義有限差分法 22
第四章 數值結果與比較 33
4.1 二維自然熱對流 33
4.1.1方形流場單邊熱源問題 33
4.1.2方形流場有一圓形熱源問題 35
4.1.3方形流場左下角有一方形熱源問題 37
4.2 三維自然熱對流 40
4.2.1正六面體單面熱源問題 40
4.2.2長方形六面體單面熱源問題 44
4.2.3圓柱體底面熱源問題 48
第五章 結論與建議 53
5.1結論 53
5.2建議 54
參考文獻 55
個人簡歷 222

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