兩球沿連心線垂直方向等速運動下流體力之理論分析

Theoretical Investigation of Hydrodynamic Force when Dual Spheres Move Constantly and Perpendicularly to the Line of Centers

陳冠宇(Kuan-Yu Chen)

指導教授 ： 楊馥菱

國立臺灣大學/工學院/機械工程學研究所/碩士(2017年)

https://doi.org/10.6342/NTU201703952

摘要

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。

關鍵字

勢流理論 ； 邊界層理論 ； 阻力係數

並列摘要

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.

並列關鍵字

two spheres ； potential flow ； boundary layer ； drag coefficient

參考文獻

[1] A. B. Basset. On the motion of two spheres in a liquid, and allied problems. Proceedings of the London Mathematical Society, 18:369–377, 1887.

[2] R. Chen and Y. Lu. The flow characteristics of an interactive particle at low reynolds number. International Journal of Multiphase Flow, 25:1645–1655, 1999.

[4] R. Folkersma et al. Hydrodynamic interactions between two identical spheres held fixed side by side against a uniform stream directed perpendicular to the line connecting the spheres’ centres. International Journal of Multiphase Flow,26:877–887, 2000.

[6] I. Kim et al. Three-dimensional flow over two spheres placed side by side. Journal of Fluid Mechanics, 246:465–488, 1993.

[9] T. Miloh. Hydrodynamics of deformable contiguous spherical shapes in an incompressible inviscid fluid. Journal of Engineering Mathematics, 11(4):349–372, 1977.