本研究針對計算流體力學（Computational Fluid Dynamics, CFD）領域進行數值模擬分析，旨在應用有限元素法（Finite Element Method, FEM）求解流體與熱傳守恆方程式，涵蓋質量守恆、動量守恆及能量守恆等三大基本方程。此類守恆式的數值求解對於解析流體運動與熱傳輸相關工程問題具有關鍵意義。
研究首先選擇適當之測試案例，推導其解析解，並建立以有限元素法為基礎之數值模擬程式，透過與解析解之誤差分析驗證數值方法之精度與穩定性。接著，以經典Lid-Driven Cavity問題作為基準案例，與文獻結果進行比對，以檢驗本研究模型在流體行為再現及流場結構預測上的可靠性。最後，將自建程式所得之模擬結果與商業軟體Simcenter STAR-CCM+進行比較，以進一步評估本方法於物理合理性與工程應用層面的可行性。
模擬結果顯示，在適當的網格分佈與邊界條件設定下，有限元素法所得之結果能準確再現經典剪力驅動迴流結構，並於速度與溫度場分佈上呈現與文獻及商用軟體一致之趨勢。雖局部區域仍存在一定差異，然整體物理場分佈合理且穩定，顯示本研究所建構之數值模型具良好準確性與可擴展性。綜上所述，本研究建立之有限元素模擬架構具備穩定性與合理性，可為後續數值方法改良及簡單熱流耦合應用提供初步參考。

This thesis presents the development of a self-implemented finite element method (FEM) solver for incompressible flow coupled with heat transfer. The solver is constructed from the governing equations to the numerical implementation, allowing explicit control over spatial discretization, time integration, and pressure–velocity coupling. The main objective is not only to obtain simulation results, but also to clearly examine the numerical behavior and applicability of the adopted schemes. The solver is assessed through three levels of verification and validation: (i) numerical verification using manufactured analytical solutions with systematic grid refinement, (ii) benchmark validation using the classical lid-driven cavity flow at Reynolds numbers Re = 100, 400, and 1000, and (iii) cross-comparison with the commercial CFD software Simcenter STAR-CCM+ under matched geometry, mesh resolution, boundary conditions, and time-step settings. The benchmark results show that the present solver accurately reproduces the primary vortex and corner eddies in the lid-driven cavity, and the centerline velocity profiles agree well with reference data from the literature. Manufactured-solution tests confirm the expected spatial convergence behavior, providing quantitative evidence of the numerical correctness of the implementation. In the coupled thermo-fluid cavity comparisons with STAR-CCM+, strong agreement is observed at low Reynolds number in both flow and temperature fields, including overall distributions and monitoring-line profiles. As the Reynolds number increases, discrepancies gradually appear in localized extrema and near-wall gradient regions, reflecting intrinsic limitations of the current numerical strategy under convection-dominated conditions. These differences are mainly attributed to the accumulation of splitting errors in the fractional-step pressure–velocity coupling and the increased sensitivity of second-order Adams–Bashforth extrapolation for convective terms at higher Reynolds numbers. Overall, the developed FEM solver demonstrates stable numerical behavior, reliable reproduction of key physical features, and consistent convergence properties for low-to-moderate Reynolds-number thermo-fluid problems. The present work also clarifies the solver’s current applicability range and provides a clear basis for future improvements targeting more strongly convection-dominated and multiphysics simulations.

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