以力元理論分析在低雷諾數下有限翼之非定常氣動力特性

Abstract

本論文主要是以力元理論來分析低雷諾數下有限翼之非定常氣動力特性。在文中探討的雷諾數為100與300，物體外型則有平板與三角翼兩種。其中考慮的平板其展弦比(AR)與攻角(AoA)分別為AR=1、2和3，AoA=5~60度之間，而三角翼的設定則有AR=1、2和4，AoA=15、30、45三個角度。張建成教授及其研究群所發展之力元理論提供了我們一個方法來解析流場中非零渦度流元與翼板受力之間的關係。因此在本論文中，我們透過力元理論分別解析有限平板的前緣渦、翼尖渦、後緣渦與表面渦度，以及三角翼的前緣渦、後緣渦、表面渦度與物體升、阻力之間的關係。與傳統壓力積分法(pressure force analysis, PFA)不同，力元理論的觀點是透過流元的渦度與物體受力作連結(vorticity force analysis, VFA)。在本文中透過比較VFA與PFA在不同截面之間的差異得到許多有趣的結果。 (1) 在平板的例子中，我們透過數個垂直於翼展方向的平面將整個流域切割成數個截面，接著對每個截面分別求其VFA與PFA的貢獻，藉由比較兩著分布結果可當作流場三維性的指標。像是高展弦比(AR=3)的平板，其PFA與VFA在平板中線區域差異非常小。反之低展弦比(AR=1)的平板，PFA在平板上方大部分的區域都明顯的大於VFA。而PFA與VFA之間的差距會在兩翼截面以及外側區域被補足。通過進一步分解渦度貢獻到縱向渦度(在二維流場中唯一的渦度分量)和其他兩個正交的橫向渦度，我們發現這兩類渦度貢獻的消長是流場三維性的指標之一。 (2) 在三角翼的例子中，我們透過兩種不同的切割方法來分析體渦度在截面上的貢獻，並且藉由物理的觀點來分析得到的結果。這兩種截面分別為垂直於入流方向的截面與由三角翼頂點為原點切割而成的扇形截面。我們分別對這兩種不同的切割截面計算其PFA以及其對應的VFA貢獻。由第一個方法可以觀察到PFA與VFA的差異在三角翼前緣並不顯著，這是因為當渦漩結構緊貼附於三角翼表面時，兩者之間差異會減小。同樣的，我們也將渦度貢獻分解成縱向渦度貢獻和橫向渦度貢獻兩個部分，透過這樣的分解可以讓我們更清楚的看出各渦度的貢獻量。我們發現在靠近翼板頂點的截面其縱向渦度貢獻較大，但是在靠近翼板後緣的截面其貢獻則會減小，甚至提供負升力貢獻。而橫向渦度的貢獻則剛好相反，在靠近頂點的截面貢獻較少，但是在靠近後緣的截面貢獻較大。而第二種切割方法所獲得的結果可以與主要的升力來源互相對照參考。由結果可發現貼附於三角翼中央的再附著區其升力貢獻主要是來自於縱向渦度。而位於翼板上方中央兩側提供負升力的再回流區其貢獻主要來自於橫向渦度。剪切區則與前面兩個區域不同，其貢獻來源包含了縱向渦度與橫向渦度。除此之外，我們也注意到三角翼板外側區域的貢獻遠低於有限平板的外側區域貢獻。其原因為三角翼兩側的前緣渦會從兩翼往翼板中心方向發展，而有限平板的翼尖渦主要是向上方發展。雖然三角翼與有限平板在流場穩定後的升力差不多，但三角翼在啟動後的瞬間升力會比平板高。由力元理論的觀點來看，三角翼橫向渦度的貢獻明顯大於有限平板，但是縱向渦度還是三角翼主要的升力來源。總結來說，力元理論提供了一種更好的物理概念來連結物體受力與翼板附近渦漩結構之間的關係。

In this study, we consider various contributions to the forces on an impulsively started finite plate or delta wing from the perspective of a diagnostic vorticity force element theory. The plate has an aspect ratio (AR) between 1 and 3, and is placed at different angles of attack (AoAs=5o、10o、15o、30o、45o、60o), while the Reynolds number is either 100 or 300. In contrast, the delta wing has an AR equal to 1, 2 and 4, and is placed at three different AoAs (15o、30o、45o) , while the Reynolds number is fixed at 300. The theory enables us to quantify the contributions to the forces exerted on the wing in terms of fluid elements with non-zero vorticity, such as in the tip vortices (TiVs), leading- and trailing-edge vortices (LEVs and TEVs) of the plate, and in the primary vortex、LEV and TEV of the delta wing. This line of force analysis has been pursued for two-dimensional flow in the previous studies by Chang’s research group. In contrast to the pressure force analysis (PFA), the vorticity force analysis (VFA) reveals new salient features in its applications to three-dimensional flow by examining sectional force contributions. (1) For the plate cases, we divide the whole flow space to several sections by a number of parallel planes perpendicular to the spanwise direction, and evaluate the pressure force and the corresponding vorticity force in each section. It is found that at a large aspect ratio (AR=3), the force distributions of PFA and VFA show close agreements with each other in the middle sections, while at a lower aspect ratio (AR=1), the force distribution of PFA is substantially higher than that of VFA in most of the sections. The difference is compensated for by the contributions partly by the edge sections and mainly by the vortices in the outer regions. It is also revealing to decompose the vorticity into the spanwise (longitudinal) component (LC-the only one in two-dimensional flow) and the other two orthogonal (transverse) components (TCs). The relative importance of the force contributions credited to the TCs in the entire flow regions as well as in the two outer regions signifies the three-dimensional nature of the flow over a finite plate. (2) For the delta wing cases, we divide the whole flow domain by vertical planes in two ways to distinguish the force distributions between PFA and VFA. (i) First, the whole space flow is divided to several sections by a number of parallel planes perpendicular to the spanwise direction, and evaluate the pressure force and the corresponding vorticity force in each section. The deviation between the PFA and VFA is found to be small at sections near the apex due to the LEVs being attached closely to the delta wing surface. The LC contributes significantly to the lift at the sections near the apex, but the contribution decreases or even turns to be negative at the sections near the trailing edge. Besides, we found that the contribution from the TCs at the sections near the apex is not significant, but the lift contribution increases at the sections near the trailing edge. (ii) Second, we divide the whole flow field by a number of vertical planes emanating from the apex of delta wing, and evaluate the pressure force and the corresponding vorticity force in each section. We find that the positive lift source from the region of re-attachment at the middle section is mainly due to the LC, and that the negative lift from the region of re-circulation region is due to the TCs. The shear layers, containing transverse and longitudinal components, provide positive lift contribution. It is also observed that the contributions from the two outer regions of the delta wing are less than those of the finite plate. This is because the LEVs of the delta wing tend to be attached to the wing by bending themselves toward the central section, while the TiVs of the plate wing evolve more upwards. Finally, we compare the lift forces of the finite plate and delta wing from the vorticity force viewpoint. The delta wing acquires a higher lift in a time period after the impulsive start, while the lift coefficients of the delta wing and finite plate make little difference in the later stages of flow development. The delta wing is shown to have larger contributions from the transverse components than the finite-plate wing, yet the main source of lift elements come from the longitudinal component. The present VFA provides a better understanding by relating the forces directly to the various sources of vorticity (or vortex structures) on or near the wings.