鯊魚表皮皮齒之特殊結構被認為是鯊魚能夠擁有高流速的主要原因之一，其流體流經皮齒之流場能展示其降阻能力。為了調查鯊魚皮齒結構中流體行為與阻力在不同雷諾數下的變化情形，本研究透過使用開源有限元素平台FEniCS來模擬納維爾史托克(Navier-Stokes)方程式。在FEniCS平台下，我們可以便捷的實施伽遼金有限元素法(Galerkin Finite Element Method)並引入多尺度變分法(Variational Multiscale Method)來穩定方程式中的平流項(Advection Term)，同時多尺度變分架構也提供了渦流模擬所需之細節項。目前本研究針對光滑球體、鯊魚仿生單一皮齒結構於不同流場下進行阻力分析與參數化研究，接著針對真實鯊魚皮齒結構—燈籠鯊(Lantern Shark)、長鬚棘鮫(Mandarin Dogfish Shark)、尖吻鯖鯊(Mako Shark)等三種鯊魚之鯊魚皮齒進行流場參數模擬，探討各個參數對於固體表面減阻效果的不同，作為未來進行鯊魚表皮阻力分析的參照資料；本文最後章節初步探討流固耦合基礎架構分析提供給作為開源有限元素法軟體FEniCS使用者未來的研究參考。

The unique geometry of the microstructure on the shark-skin surface, known as denticles, is the main reason sharks can swim at high speed. In order to discover the relationship between the complicated denticle structures with the associated flow pattern, we perform a series of parameter studies, including the drag force analysis and flow stability test under different denticle geometry. In this research, the incompressible Navier-Stokes (N-S) equations are chosen as the fluid model; the Galerkin finite element method with variational multiscale method (VMS) is adapted. The VMS aims to stabilize the instability due to the advection in the N-S equations, as well as introduces the fine-scale terms that act as the large eddy simulation (LES) in the fluid model. The framework was implemented under the open-source finite element platform, called FEniCS. We first perform the numerical testing on the flow over a sphere and a bionic shark-skin structure to investigate the flow pattern under different Reynolds number, which validates the effectiveness of the proposed framework. Then, a parameter study is employed for different shark species with associated unique shark denticle geometry. The drag-reduction effects among different shark-skin are revealed, and can be used as a reference for future design on the bio-inspired structure. Finally, the last chapter preliminarily discusses the analysis of the fluid-structure interaction within FEniCS.

摘要 i
ABSTRACT ii
目錄 iv
圖目錄 vii
第一章 緒論 1
1.1 前言 1
1.2 流體阻力之成形原因 4
1.3 文獻回顧 4
1.3.1 仿生結構表面流體研究分析概況 5
1.3.2 仿生減阻研究與應用 12
1.4 研究動機 14
第二章 基礎理論 16
2.1 納維爾史托克(NS)方程和雷諾數(Re)的影響 16
2.2 NS方程多尺度變分穩定有限元方法(VMS-FEM) 17
2.3 數值流程 20
第三章 鯊魚皮齒結構數值模型 22
3.1 真實鯊魚皮齒結構 22
3.1.1 燈籠鯊(Lantern Shark) 24
3.1.2 長鬚棘鮫(Mandarin Dogfish Shark) 24
3.1.3 尖吻鯖鯊(Mako Shark) 25
3.2 數值設定 26
3.2.1 邊界條件 26
3.2.2 阻力以及阻力係數計算方式 28
第四章 皮齒結構流場特性分析 29
4.1 幾何模型阻力驗證 29
4.1.1 光滑球體流場幾何模型 29
4.1.2 光滑球體流場阻力分析 30
4.2 鯊魚流場阻力分析 34
4.2.1 鯊魚仿生單一皮齒結構流場阻力分析 36
4.2.2 真實鯊魚皮齒結構流場阻力分析 40
4.2.2.1 燈籠鯊(Lantern Shark) 40
4.2.2.2 長鬚棘鮫(Mandarin Shark) 42
4.2.2.3 尖吻鯖鯊(Mako Shark) 44
4.2.3 流體控制穩定性分析 46
第五章 流固耦合模擬架構開發 49
5.1 流固耦合模擬簡介 49
5.1.1 任意拉格朗日-歐拉描述法簡介 50
5.1.2 流固耦合統御方程式 52
5.1.2.1 拉格朗日結構方程式 52
5.1.2.2 歐拉流體方程式 54
5.2 流固耦合數值案例 55
5.2.1 幾何模型 55
5.2.2 結果與討論 57
第六章 結論與未來研究方向 60
6.1 結論 60
6.2 未來研究方向 61
參考文獻 62
附錄A(Appendix A) 67
附錄B(Appendix B) 76

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