旋轉球體在運動過程中，其運動狀態各不相同，運動速度以及旋轉速度各不相同，應瞭解旋轉球體之旋轉速度大小與方向性。球體旋轉通常分為左旋、側旋、上旋、下旋、順旋以及逆旋，但在旋轉球體運動過程中的旋轉通常是以上六大旋轉中多種旋轉綜合作用。本論文研究係採用計算流體力學(computational fluid dynamics, 簡稱CFD)方法獲取旋轉球體在不同運動狀態下的流場及運動軌跡，與運動學求解方法進行比較驗證CFD方法的優勢。本論文研究採用CFD數值方法對旋轉球體在不同轉速大小方向運動進行模擬計算，對比分析不同方法對於旋轉球體之流場特性與運動軌跡的影響。吾人由結果得知：基於CFD方法的計算結果比運動學求解方法更符合實際；旋轉方向對旋轉球體的運動軌跡有很大影響，上旋球軌跡偏低，下旋球軌跡偏高，側旋球軌跡向左向偏轉；旋轉速度對旋轉球體的軌跡也有很大影響，上旋球球體的旋轉速度越大其落點距離越近；球體的旋轉運動將導致更為複雜的流場狀態。而對於旋轉球體運動的流場特性模擬與分析可知：相比於無旋轉球體，旋轉球體之流場分佈較為複雜，無旋轉球體速度及壓力流場具有對稱性，旋轉球體因受旋轉作用其流場不具有對稱性。本文研究結果可提供運動選手(如棒球、棒球球體等)在訓練過程中，可藉此找出更新穎的變化球種、打擊時機點或球路判斷點，提高選手掌握球體擊球點軌跡可控性的參考與依據。

During the movement of the rotating sphere, depending on its motion state, movement speed and rotation speed, the rotation speed and direction of the rotating sphere should be known in detail. The rotation of the sphere is usually divided into left-handed, side-pinned, up-spindle, down-spin, spin-spin, and counter-rotation, but the rotation during the rotation of the rotating sphere is usually a combination of multiple rotations in the above six major rotations. In this thesis, the computational fluid dynamics (CFD) method is used to obtain the flow field and motion trajectory of the rotating sphere under different motion states. The advantages of the CFD method are verified by comparison with the kinematic solution method. In this thesis, the CFD numerical method is used to simulate the motion of rotating spheres in different rotational speeds. The effects of different methods on the flow field characteristics and motion trajectories of rotating spheres are compared and analyzed. From the results, we know that the calculation results based on the CFD method are more realistic than the kinematics solution method; the rotation direction has a significant influence on the trajectory of the rotating sphere. The upper spiral track is low, the lower spin track is high, and the side spin track is deflected to the left. The rotational speed also has a significant influence on the trajectory of the rotating sphere. The higher the rotational speed of the topspin sphere, the closer the falling point distance; the rotational motion of the sphere will lead to a more complicated flow field state. For the simulation and analysis of the flow field characteristics of the rotating sphere, it can be seen that the flow field distribution of the rotating sphere is more complicated than that of the non-rotating sphere. There is no symmetry of the rotating sphere velocity and the pressure flow field, and the rotating sphere is subjected to the unsymmetrical rotating flow field. The results of this research can provide sports players in the training process, which can be used to find newer changes in the ball type, hitting the timing point or the ball path judgment point, and improve the player's grasp of the ball hitting point track.

目錄
誌謝 I
中文摘要 II
Abstract III
目錄 IV
圖目錄 VII
表目錄 XII
符號索引 XIII
第一章、前言 1
1.1 研究背景 1
1.1.1 棒球的歷史演進 2
1.1.2 硬式棒球的由來 3
1.1.3 各式球種的由來 4
1.2 文獻回顧 1
1.2.1 作用在飛行棒球上之力的來源 2
1.2.2 阻力 3
1.2.3 升力 4
1.2.4 尾流 8
1.2.4 風洞實驗 10
1.3 研究方法 1
1.4 研究內容與大綱 1
第二章、球體的結構 3
2.1 棒球結構 3
2.2 球皮粗細與飛行距離 3
2.3 幾何模型與網格系統 4
2.4 飛行軌跡探討相關文獻 7
第三章、數學模式 9
3.1 統御方程式 9
3.2 紊流模式 11
3.2.1 大渦紊流模型(Large Eddy Simulation, LES) 12
3.2.2 紊流傳遞模式(Turbulent Transport Models, TTM) 13
3.3 數值模式 14
3.3.1 對流項 15
3.3.2 平行運算 15
3.4 邊界條件 16
3.5 模擬分析流程 17
3.6 程式操作設定 20
第四章、模擬結果與討論 22
4.1 三維球體與網格無關測試 22
4.2 三維球體二縫線分析 24
4.3 三維球體四縫線分析 32
4.4 三維球體旋轉分析 40
4.4 尾流對運動軌跡的影響 47
第五章、結論與建議 53
5.1 結論 54
5.2 建議 56
參考文獻 57


圖目錄
圖1.1 作用在飛行棒球上的力 4
圖1.2 變化球(a)扣手腕的動作；(b)球體上表面的氣流 5
圖1.3 變化球(a)直球的尾勁；(b)垂直於運動方向的側力及(c)不對稱剝離球體 5
圖1.4球體周圍之邊界層性質 (資料來源：Anderson, 1991) 2
圖1.5 棒球飛行時，其球面上所受到的壓力與剪應力 3
圖1.6 不同雷諾數下的阻力係數 (資料來源: Landau &Lifshitz, Fluid Mechanics) 6
圖1.7 雷諾數與阻力係數之關係 (資料來源: Morrison 2013) 6
圖1.8 從打擊者方向看過去的各種右投手變化球的旋轉方向 6
圖1.9 馬格那斯力示意圖 (資料來源：Susan[20], 2003) 7
圖1.10 球體因旋轉方式Top-spin(旋轉前進)和Back-spin(逆旋)而產生不同方向之力(資料來源：Susan[20], 2003) 7
圖1.11 圓柱旋轉落水產生的馬格那斯效應 (資料來源：Hermans[21], 2005) 8
圖1.12 不同角速度下球體的升力係數與雷諾數之關係 (資料來源: Oesterléet al., 1998) 8
圖1.13 尾流的型態變化圖 (資料來源: Thompson et al., 2001) 9
圖1.14 在不同Re下，通過圓球所產生尾流的流線(資料來源: Taneda, 1956) 9
圖1.15 (a)雷諾數400及450時阻力係數有週期性變化的時序圖；(b)尾流分離角度與雷諾數之關係(資料來源: Lee, 2000) 10
圖1.16 風洞實驗棒球飛行 11
圖1.17 風洞實驗中的棒球飛行 (資料來源：Am.J.Phys.27, p589-596, 1959) 11
圖1.18 風洞實驗攝影：(a)網球在旋轉情形下之風洞實驗(Goodwill[40], 2004)；(b)圓柱障礙物之煙霧擴散流場(Mavroidis[41], 2003) 12
圖2.1 球皮粗細與飛行距離 4
圖2.2 棒球的幾何模型及尺寸 5
圖2.3二維複合式網格 6
圖2.4 三維網格系統 6
圖2.5 三維非結構式網格計算域 7
圖2.6 三維結構式網格計算域 7
圖2.7 軟式棒球與硬式棒球 8
圖3.1 多顆CPU之平行運算系流示意圖 16
圖3.2 格點計算域分割示意圖 16
圖3.3 邊界條件及計算域尺寸示意圖(D=42.6 mm) 17
圖3.4 基本計算程序結構示意圖 19
圖3.5 結構化網格及座標系統示意圖 21
圖3.6 非結構化網格及座標系統示意圖 21
圖4.1 邊界條件及計算域尺寸示意圖(D=42.6 mm) 23
圖4.2 與網格數量無關之適當性比較 24
圖4.3 三維圓球二縫線模擬結果之壓力pressure等高線分佈(時間t＝0.24 sec) 24
圖4.4 三維圓球二縫線模擬結果之streamstace分佈(時間t＝0.24 sec) 25
圖4.5 三維圓球二縫線模擬結果之速度velocity等高線分佈(時間t＝0.24 sec) 25
圖4.6 三維圓球二縫線模擬結果之vorticity分佈(時間t＝0.24 sec) 25
圖4.7 三維圓球二縫線模擬結果之壓力pressure等高線分佈(時間t＝0.45 sec) 26
圖4.8 三維圓球二縫線模擬結果之streamstace分佈(時間t＝0.45 sec) 26
圖4.9 三維圓球二縫線模擬結果之速度velocity等高線分佈(時間t＝0.45 sec) 26
圖4.10 三維圓球二縫線模擬結果之vorticity分佈(時間t＝0.45 sec) 27
圖4.11 三維圓球二縫線模擬結果之壓力pressure等高線分佈(時間t＝0.6 sec) 27
圖4.12 三維圓球二縫線模擬結果之streamstace分佈(時間t＝0.6 sec) 28
圖4.13 三維圓球二縫線模擬結果之速度velocity等高線分佈(時間t＝0.6 sec) 28
圖4.14 三維圓球二縫線模擬結果之vorticity分佈(時間t＝0.6 sec) 29
圖4.15 維圓球二縫線模擬結果之壓力pressure等高線分佈 29
圖4.16 三維圓球二縫線模擬結果之streamstace分佈 30
圖4.17 三維圓球二縫線模擬結果之速度velocity等高線分佈 30
圖4.18 三維圓球二縫線模擬結果之vorticity分佈 30
圖4.19 三維圓球二縫線之阻力分佈(drag)與隨時間變化之頻率大小(amplitude) 31
圖4.20 三維圓球二縫線之升力分佈(lift)與隨時間變化之頻率大小(amplitude) 31
圖4.21 三維圓球四縫線模擬結果之壓力pressure等高線分佈(時間t＝0.42 sec) 33
圖4.22 三維圓球四縫線模擬結果之streamstace分佈(時間t＝0.42 sec) 33
圖4.23 三維圓球四縫線模擬結果之速度velocity等高線分佈(時間t＝0.42 sec) 34
圖4.24 三維圓球四縫線模擬結果之vorticity分佈(時間t＝0.42 sec) 34
圖4.25 三維圓球四縫線模擬結果之壓力pressure等高線分佈(時間t＝0.54 sec) 35
圖4.26 三維圓球四縫線模擬結果之streamstace分佈(時間t＝0.54 sec) 35
圖4.27 三維圓球四縫線模擬結果之速度velocity等高線分佈(時間t＝0.54 sec) 35
圖4.28 三維圓球四縫線模擬結果之vorticity分佈(時間t＝0.54 sec) 36
圖4.29 三維圓球四縫線模擬結果之壓力pressure等高線分佈(時間t＝0.6 sec) 37
圖4.30 三維圓球四縫線模擬結果之streamstace分佈(時間t＝0.6 sec) 37
圖4.31 三維圓球四縫線模擬結果之速度velocity等高線分佈(時間t＝0.6 sec) 38
圖4.32 三維圓球四縫線模擬結果之vorticity分佈(時間t＝0.6 sec) 38
圖4.33 三維圓球四縫線之阻力分佈(drag)與隨時間變化之頻率大小(amplitude) 39
圖4.34 三維圓球四縫線之升力分佈(lift)與隨時間變化之頻率大小(amplitude) 39
圖4.35 三維圓球四縫線之Q-criterion分佈 39
圖4.36 三維圓球施以旋轉外力模擬結果之壓力pressure等高線分佈(時間t＝0.42 sec) 40
圖4.37 三維圓球施以旋轉外力模擬結果之streamstace分佈(時間t＝0.42 sec) 41
圖4.38 三維圓球施以旋轉外力模擬結果之速度velocity等高線分佈(時間t＝0.42 sec) 41
圖4.39 三維圓球施以旋轉外力模擬結果之vorticity分佈(時間t＝0.42 sec) 42
圖4.40 三維圓球施以旋轉外力模擬結果之壓力pressure等高線分佈(時間t＝0.54 sec) 43
圖4.41 三維圓球施以旋轉外力模擬結果之streamstace分佈(時間t＝0.54 sec) 43
圖4.42 三維圓球施以旋轉外力模擬結果之速度velocity等高線分佈(時間t＝0.54 sec) 43
圖4.43 三維圓球施以旋轉外力模擬結果之vorticity分佈(時間t＝0.54 sec) 44
圖4.44 三維圓球施以旋轉外力模擬結果之壓力pressure等高線分佈(時間t＝0.6 sec) 44
圖4.45 三維圓球施以旋轉外力模擬結果之streamstace分佈(時間t＝0.6 sec) 44
圖4.46 三維圓球施以旋轉外力模擬結果之速度velocity等高線分佈(時間t＝0.6 sec) 45
圖4.47 三維圓球施以旋轉外力模擬結果之vorticity分佈(時間t＝0.6 sec) 45
圖4.48 三維圓球四縫線之阻力分佈(drag)與隨時間變化之頻率大小(amplitude) 46
圖4.49 三維圓球四縫線之升力分佈(lift)與隨時間變化之頻率大小(amplitude) 46
圖4.50 三維圓球軸心(axis)模擬結果之壓力pressure等高線分佈(時間t＝0.42 sec) 48
圖4.51 三維圓球軸心(axis)模擬結果之streamstace分佈(時間t＝0.42 sec) 48
圖4.52 三維圓球軸心(axis)模擬結果之速度velocity等高線分佈(時間t＝0.42 sec) 48
圖4.53 三維圓球軸心(axis)模擬結果之vorticity分佈(時間t＝0.42 sec) 48
圖4.54 三維圓球軸心(axis)模擬結果之壓力pressure等高線分佈(時間t＝0.54 sec) 49
圖4.55 三維圓球軸心(axis)模擬結果之streamstace分佈(時間t＝0.54 sec) 49
圖4.56 三維圓球軸心(axis)模擬結果之速度velocity等高線分佈(時間t＝0.54 sec) 49
圖4.57 三維圓球軸心(axis)模擬結果之vorticity分佈(時間t＝0.54 sec) 50
圖4.58 三維圓球軸心(axis)模擬結果之壓力pressure等高線分佈(時間t＝0.6 sec) 50
圖4.59 三維圓球軸心(axis)模擬結果之streamstace分佈(時間t＝0.6 sec) 50
圖4.60 三維圓球軸心(axis)模擬結果之速度velocity等高線分佈(時間t＝0.6 sec) 51
圖4.61 三維圓球軸心(axis)模擬結果之vorticity分佈(時間t＝0.6 sec) 51
圖4.62 三維圓球軸心(axis)之阻力分佈(drag)與隨時間變化之頻率大小(amplitude) 51
圖4.63 三維圓球軸心(axis)之升力分佈(lift)與隨時間變化之頻率大小(amplitude) 52


表目錄
表1.1 不同年代學者的臨界雷諾數及尾流現象的研究結果 10
表3.1 數值模擬解法設定 18
表3.2 計算流體力學商業軟體 19

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