平板附近渦流對升力的影響

Abstract

本文旨在利用二維勢流（potential flow）理論，來建構一個具有渦流之平板流場，以大致模擬及定性分析一平板翼在流場中產生渦流時之升力變化情形。本文的方法是先利用均勻流（uniform flow）、渦流（vortex）、偶流（doublet）的複勢（complex potential）先建立一個圓柱流場，再使用Joukowski轉換得到平板流場，然後以Kutta condition修正後端銳緣速度無限大的問題，最後使用Blasius積分求出平板升力之公式，而計算升力值與繪製流場流線與向量圖的工作將交由電腦程式（Fortran及Tecplot）來執行。 當流場中只有一個渦流且位置在平板上方時，若固定水平方向位置並改變垂直方向位置，我們可以發現此渦流越靠近平板上緣，將產生越大的升力，與平板距離超過某範圍後，將產生負的升力；若固定垂直方向位置並改變水平方向位置，升力極大值將發生在平板後緣附近。 渦流在某些強度並放置在某些區域時，會較無渦流時有較高的升力。當改變渦流強度時，若無攻角則無明顯變化，僅在靠近平板上緣或後緣附近以後的區域，會產生較無渦流時更大的升力；若有攻角時將平板產生額外的升力，則渦流強度超過一定的大小之後白努利效應才開始明顯，最後同樣只有在靠近平板上緣與後緣附近之後的部分能產生較大的升力。 最後，本文由延伸的Lagally定理發展出具有多個渦流的平板流場通解，即可利用此通解輔以電腦程式來繪製出任何需要的相關流場。

This article aims to use two-dimensional potential flow theory, to construct a flow field that uniform flow past through a flat plate which has vortices above the plate, so we can simulate and qualitative analyze the variations of lift when vortices generated in the flow field of a flat plate wing. The method in this article is to first use uniform flow, vortex, doublet to establish the complex potential of a circular cylinder flow field, and then map it into a plate flow field by using Joukowski transformation, then use Kutta condition to amend the problem of infinite velocity at the sharp trailing edge, finally we can obtain the lift of the plate by using the Blasius’ integral laws. While the calculating values of lift and drawing the streamlines and the vector filed will be handed over by computer programming (Fortran and Tecplot). When there is only one vortex above the flat plate, if we fix the horizontal position and change the vertical position, it will produce greater lift if the vortex is close to the plate enough, and it will produce negative lift after a certain distance away from the plate; if we fix the vertical position and change the horizontal position, the maximum lift will occur near the trailing edge at the plate. The vortex will result in higher lift than no vortex when the vortex is been placed in some areas. When changing the circulation strength of the vortex, if there’s no angle of attack, only near the plate’s upper surface, or near and after the trailing edge region, will produce greater lift than there’s no vortex; if there exists an angle of attack, it will generate additional lift, then the circulation must be over a certain strength so that the Bernoulli effect may be significant, only near the plate’s upper surface, or near and after the trailing edge region will produce the larger lift, too. Finally, the general solution of the flow field of the plate with multiple vortices will be developed by using the extension of Lagally theorem, we can use this general solution accompanied by a computer program to map out any relevant required flow.