低黏滯性流體中二正向靠近球體受力之理論推導

Abstract

本論文以理論推導的方式探討兩顆任意大小的球體以任意大小的速度正向靠近時所受到之總阻力。本論文大致上分為兩的部分：位勢流理論和邊界層理論。在位勢流理論的部分，我們主要討論兩任意大小之球體，在一無限大且理想的流場中以任意大小的速度正向靠近。由於理想流流場中不存在黏滯力，故球體在運動過程中所受到唯一的阻力將來自於流體施予球體的正向壓力。傳統的研究為一顆球體在一理想流流場中運動，唯有其運動方式為非穩態時，該球體才會受到流場所施予的阻力。本篇論文將藉由球面積分，得到某一球在運動過程中所受到的阻力，我們稱此力為形狀阻力(form drag)。和傳統結果不同的是，即使球體是以等速度的方式在運動仍然會受到阻力，其原因為流場中存在了其他的固體邊界而導致壓力場的改變。 在邊界層理論的部分，我們延續在位勢流理論的問題設定，但是不同的是我們討論的球體不再是任意大小，而是兩個相同大小的球體，以相同的速度正向靠近。此外，我們假設雷諾數為1000。經過數值方法的計算，我們可以求出此流場在某一球體上所施加的摩擦阻力。在邊界層中，我們假設流場的壓力不會隨著垂直固體邊界的方向變化，故我們可以由邊界層外的壓力場推測邊界層內的壓力場。於是，在我們所關心的問題中球體所受到的總阻力可以經由形狀阻力和摩擦阻力的相加所得到。

This thesis investigates the hydrodynamic force for two immersed sphere approaching each other via theoretical analysis. The particle Reynolds number is assume to be high enough so that the boundary layer equation was solved in junction with the corresponding potential flow theory for the outer invisid flow. The potential flow when two immersed spheres of arbitrary size move along the line of centers at arbitrary velocity is solved first and a general formula for the pressure drag is developed. We then focus on the special case when the two spheres are identical and at the same approaching velocity. The inviscid pressure field is developed first and integrated into the boundary layer equation to solve the wall shear stress. Both the pressure and the shear stress were integrated along the sphere surface to obtain a pressure and a friction drag which were summed to give the total drag. Both force components increase monotonically with diminishing inter-sphere separation and hence the same trend is found for the total drag. Further, it is also found that pressure drag dominates the total drag in far field but becomes a minor role in contributing to the total drag when friction drag rise dramatically at small separations.