臺灣博碩士論文加值系統

本論文結合勢流理論與邊界層理論來探討兩顆不可變形的球在無限大流域中以等速沿垂直球心連線方向的運動時，不同間距下球所受到的力。首先用勢流理論推導出在不考慮黏性的情況下球的形狀阻力，結果顯示兩球會互相吸引，另外還可以觀察到停滯點因為第二顆球的存在而從球的最前端和最後端往兩球間空隙偏移。接著將勢流理論求得的壓力場，與邊界層理論結合來計算球所受到的摩擦阻力，進而與形狀阻力相加求得總阻力。由於流場的不對稱性，本論文提出一個基於 Blasius series method 的近似方法來估計總阻力係數。此方法計算結果顯示在單球雷諾數為 100 時，阻力係數會隨兩球間距的增加而下降，這點和許多實驗與數值模擬觀察到的結果一致，不過計算出的阻力係數的值是有差距的，這方面的問題將會在最後作探討。

This thesis integrates potential flow theory and boundary layer theory to derive an approximate flow solution when two rigid spheres move constantly and perpendicularly to the line connecting their centers in an incompressible fluid of infinite extent. Particular interest is to derive the hydrodynamic force when the two spheres are kept at various separation distances. Potential flow theory is first applied to derive the inviscid force experienced by the spheres and the spheres are shown to attract each other when moving side-by-side. Furthermore, the phenomenon of the stagnation points shifting away from the foremost and the rearmost points of the spheres due to the presence of a second sphere is observed. Viscous force acting on the spheres is approximated by the boundary layer theory using the pressure field solved from the inviscid problem. Due to flow asymmetry from the foremost sphere point, an approximation based on Blasius series method is proposed to estimate the drag coefficient. The approach successfully captures the trend of decreasing drag coefficient with increasing gap between the spheres, observed in experiments and simulations, at single sphere Reynolds number of 100. However, quantitative discrepancy is found and will be discussed together with the model limitations and implementation issues.

摘要 iii
Abstract v
1 Introduction 1
1.1 Background and Objective . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Inviscid Theory 5
2.1 Velocity Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Velocity Potential . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.3 Determination of Coefficients . . . . . . . . . . . . . . . . . . 9
2.2 Convergence Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Flow around the Spheres . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Tangential velocity . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.2 Stagnation Points . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1 Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.2 Comparison with Solutions in Literature . . . . . . . . . . . . 21
2.4.3 Spheres of Different Sizes . . . . . . . . . . . . . . . . . . . . 23
2.5 Inviscid Force Acting on the Spheres . . . . . . . . . . . . . . . . . . 23
2.5.1 Pressure Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5.2 Inviscid Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.6 Lift Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Viscous Theory 37
3.1 Boundary Layer Equations for Steady Axisymmetric flow over an
Axisymmetric Body . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.1 Blasius Series Method . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Viscous Force Acting on the Spheres . . . . . . . . . . . . . . . . . . 40
3.2.1 Approximation Method . . . . . . . . . . . . . . . . . . . . . 40
3.2.2 Tangential Velocity . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.3 Evaluation of Flow Asymmetry . . . . . . . . . . . . . . . . . 42
3.2.4 Friction Drag for Two Identical Spheres . . . . . . . . . . . . 43
4 Summary and Future Work 49
4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Bibliography 51

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