本文以螺槳之尾跡流為主要研究之對象，使用邊界積分法計算尋找螺槳推進時螺葉所產生之尾跡流位置。本文假設流場內之流體滿足勢流，並以邊界積分法描繪物體邊界幾何，以及使用高斯積分法對離散後的邊界積分方程式進行積分，以求解此螺槳葉片上之速度勢分布與葉片尾端之環流量，而後將葉片上之速度勢與環流量帶入邊界積分方程式，即可找到流場中各點之速度勢。本文之目標為找到螺槳之正確尾跡流位置，而所使用的方式為藉由迭代改變螺槳尾跡流曲面形狀，並且計算其法向速度，由於理論上尾跡流曲面之法向速度應為零，因此將迭代調整後的尾跡流所計算出的法向速度做平均方均跟誤差，透過迭代調整尾跡流曲面形狀將誤差值最小化，以找到最佳的尾跡流曲面位置。本文將以NDSRDC N4118螺槳為例，尋找出此螺槳所產生之尾跡流位置。

The objective of this thesis is to locate the wake position of propellers in steady flow. The flow field is assumed to satisfy on the potential theory. In addition, the Boundary Integral Equation is applied to solve the velocity potential on the boundary of the propeller’s blades. Once the strength of the velocity potential is solved, it can be substituted back into the Boundary Integral Equation to find the flow field velocity potential at all the positions. In the final, by adjusting the shape of propeller’s wake, and minimizing the normal velocity on the surface of the wake, the ultimate goal is to select an optimal position of propeller’s wake. The NDSRDC propeller N4118 are used as the test case.

口試委員會審定書 #
致謝 i
摘要 ii
ABSTRACT iii
目錄 iv
圖目錄 vii
表目錄 x
第一章 緒論 1
1.1 研究動機及背景 1
1.2 文獻回顧 2
1.3 研究目的與方法 6
第二章 基本理論 7
2.1 基本假設 7
2.2 高斯散度定理及格林定理 7
2.3 邊界積分方程式 9
2.4 格林定理之應用 12
第三章 螺槳流場數值計算 18
3.1 螺槳幾何 18
3.1.1 螺葉截面翼型幾何 20
3.1.2 螺槳軸心幾何 21
3.2 邊界積分法離散化形式 23
3.3 奇異點與近似奇異點之處理 25
3.3.1 奇異點之處理 26
3.3.2 近似奇異點之處理 27
3.3.3 尾跡流影響係數 30
3.3.4 源流積分項 32
3.4 流場速度勢及環流量計算結果 34
3.4.1 三維球體速度勢計算結果 34
3.4.2 螺槳速度勢及環流量計算結果 36
第四章 螺槳尾跡流曲面流場數值計算 38
4.1 螺葉尾跡流幾何 38
4.2 螺槳尾跡流曲面之速度勢計算 40
4.3 改變螺葉尾跡流曲面形狀與位置 43
4.3.1 調整 之葉片尖端處尾跡流流線半徑函數 44
4.3.2 調整 之葉片根部處尾跡流流線半徑函數 49
4.3.3 迭代調整葉片尖端流線與葉片根部流線 54
4.3.4 三段三次曲線近似尾跡流曲面流線之徑向螺距角 57
4.3.4 調整螺槳尾跡流流線之軸向螺距 62
4.4 數值計算結果比較與驗證 66
4.4.1 前進係數 之尾跡流位置 66
4.4.2 調整尾跡流之環流量比較 68
4.4.3 螺槳N4118 性能曲線 69
第五章 結論與展望 72
5.1 結論 72
5.2 展望 73
參考文獻 75

[1]Fredholm, E.I., “Sur une classe d''equations fonctionnelles”, Acta Math., 27, pp. 365–390, 1903.
[2]Kellogg, O. D., “Foundations of Potential Theory”, Springer, Berlin , 1900.
[3]Mikhlin, S. G., “Integral Equations”, Pergamon Press , London, 1957.
[4]Kupradze, V.D., “The method of potentials in elasticity theory”, Israel Program for Scientific Translations, Moscow, 1963.
[5]Jaswon, M. A., “Integral Equation Methods in Potential Theory─I” Proc. Roy. Soc. Lond., vol. A275, pp. 23-32, 1963.
[6]Symm, G. T., “Integral Equation Methods in Potential Theory─II” Proc. Roy. Soc. Lond., vol. A275, pp. 33-46, 1963.
[7]Lamb, H., “Hydrodynamics”, Dover, New York, 1945.
[8]Hess, J. L. and Smith, A. M., “Calculation of nonlifting potential flow about arbitrary three-dimensional smooth bodies”, J. Ship Research, vol. 7, pp. 22-44, 1964.
[9]Rizzo, F. J. “An integral equation approach to boundary value problems of classical elastostatics” Quart. Appl. Math., vol. 25, pp. 83-95, 1967.
[10]Cruse, T. A. “Numerical solutions in three dimensional elastostatics”, Int. J. Solids and Structures, vol. 5, pp. 1259-1274, 1969.
[11]Cruse, T. A. and Rizzo, F. J., Boundary Integral Equation Method, McGraw-Hill, New York, 1975.
[12]Lachat, J. C. and Watson, J. O. “A second generation boundary integral equation program for three-dimensional elastic analysis”, ASME Applied Mechanics Division National Conference, New York, 1975.
[13]Banerjee, P. K. and Butterfield R., “Boundary element method in geomechanics”, Chapter 16 in Finite element in geo-mechanics (Ed. G. Gudehus), John Wiley & Sons, New York, 1977.
[14]Brebbia, C. A., The Boundary Element Method for Engineers, Pentech Press, London, 1978.
[15]Brebbia, C. A. and Walker, S., Boundary Element Techniques in Engineering, Pentech Press, London, 1980.
[16]Brebbia, C. A., Progress in Boundary Element Method. vol. 1, Pentech Press, London, 1981.
[17]Brebbia, C. A., Progress in Boundary Element Method. vol. 2, Pentech Press, London, 1983.
[18]Brebbia, C. A., Progress in Boundary Element Method. vol. 3, Pentech Press, London, 1984.
[19]Brebbia, C. A., Progress in Boundary Element Method. vol. 4, Pentech Press, London, 1984.
[20]Brebbia, C. A., Telles, J. C. and Wrobel, L. C., “Boundary Element Techniques Theory and Applications in Engineering”, Springer Verlag, Berlin, 1984.
[21]Zang, Y. L. and Cheng Y. M., “A higher-order boundary element method for three-dimensional potential problems”, Int. J. Numer. Methods Fluid, vol. 21, pp. 321-331, 1995.
[22]Amini, S. and Wilton D. T., “An investigation of boundary element methods for the exterior acoustic problem”, Comput Methosd. Appl. Mech. Eng., vol. 54, pp. 49-65, 1986.
[23]Grilli, S. T. and Svendsen I. A., “Corner problems and global accuracy in the boundary element solution of nonlinear wave flows”, Engineering Analysis with Boundary Elements, vol. 7, pp. 178-195, 1990.
[24]Newman J. N., “Distributions od source and normal dipoles over a quadrilateral panel”, J. Eng. Math , vol. 20, pp. 113-126, 1986.
[25]Lachat, J. C. and Watson J. O., “A second generation boundary integral equation program for three-dimensional elastic analysis”, ASME Applied Mechanics Division National Conference, New York, 1975.
[26]Rizzo, F. J. and Shippy, D. J., “An advanced boundary integral equation method for three-dimensional thermo-elasticity”, Int. J. Numer. Methods Eng., vol. 11, pp. 1753-1768, 1977.
[27]Zang, Y. L. and Cheng Y. M., “A higher-order boundary element method for three-dimensional potential problems”, Int. J. Numer. Methods Fluid, vol. 21, pp. 321-331, 1995.
[28]Landweber, L. and Macagno M., “Irrotational Flow about Ship Forms”, IIHR Report, Iowa, No. 123, 1969.
[29]Webster, W. C., “The flow about arbitrary three-dimension smooth bodies”, J. Ship Research, vol. 19, pp. 206-218, 1975.
[30]Heise, U., “Numerical properties of integral equation in which the given boundary values and the solutions are defind on different curves”, Comput. Struct., vol. 8, pp. 199-205, 1978.
[31]Han, P. S. and Olson, M. S., “An adaptive boundary element method”, Int. J. Numer. Methods Eng., vol. 24, pp. 1187-1202, 1987.
[32]Johnson, R. L. and Fairweather, G., “The method of fundamental solutions for problem in potential flow”, Appl. Math Modeling, vol.8, pp. 265-270, 1984.
[33]Schulz, W. W. and Hong, S. W., “Solution of potential problems using an overdetermined complex boundary integral method”, J. Comput. Phys., vol. 84, pp. 414-440, 1989.
[34]Cao, Y. and Schultz, W. W. and Beck, R. F., “Three-dimension desingularized boundary integral methods for potential problems”, Int. J. Numer. Methods Fluid, vol. 12, pp. 785-803, 1991.
[35]Hwang, W. S., “Hypersingular boundary integral equations for exterior acoustic problems,” J. Acoust. Soc. Am., vol. 101, pp. 3336-3342, 1997.
[36]Hwang, W. S. and Huang Y. Y., “Non-singular direct formulation of boundary integral equations for potential flows,” Int. J. Numer. Mech Fluids, vol. 26, pp. 627-635, 1998.
[37]Hwang, Y. Y., “The study on potential flow by nonlinear boundary element methods,” 國立臺灣大學博士論文, 1998.
[38]Chang, J. M., “Numerical studies on desingularized Cauchy’s formula with applications to interior potential problems,” Int. J. Numer. Mech Eng., vol. 46, pp. 805-824, 1999.
[39]Yang, S. A., “On the singularities of Green’s formula and its normal derivative with an application to surface-wave-body interaction problems,” Int. J. Numer. Mech Eng., vol. 47, pp. 1841-1864, 2000.
[40]Hwang, W. S., “A boundary node method for airfoils based on the Dirichlet condition,” Comput. Methods Appl. Mech. Eng., vol. 190, pp. 1679-1688, 2000.
[41]Morino, L. and Bernardini, G., “Singularities in BIEs for the Laplace equation; Joukouski trailing-edge conjecture revisited”, Eng. Anal. Bound. Elm., vol. 25, pp. 805-818, 2001.
[42]Kerwin, J. E. and Kinnas, S. A. and Lee, J. T., and Shih, W.Z., “A surface panel method for the hydrodynamic analysis of ducted propellers”, Trans. SNAME, 95, pp. 93-112, 1987.
[43]Johnson, F.T. and Ehlers, F.E. and Rubbert, P.E., “A higher order panel method for general analysis and design applications in subsonic flow. In Proceedings of fifth International Conference on Numerical Methods in Fluid Dynamics”, Springer Verlag, 1976.
[44]Bristow, D. R. and Grose, G. G., “Modification of the douglas Neumann program to improve the efficiency of predicting component interference and high lift characteristics”, Technical Report NASA CR-3020, Langley Research Center, NASA, 1978.
[45]Morino, L. and Kuo, C. C., “Subsonic potential aerodynamic for complex configurations : A general theory”, AIAA Journal, vol 12(no.2), pp 191-197, 1974.
[46]Lee, J.T., “A potential Based Panel Method for the Analysis of Marine Propellers in Steady Flow”, PhD thesis, Massachusetts Institute of Technology, 1987.
[47]Hsin, C. Y. “Development and Analysis of Panel Methods for Propellers in Unsteady Flow”, PhD thesis, Massachusetts Institute of Technology, 1990.
[48]洪立萍，「應用邊界積分法求解二維勢流流場問題」，國立台灣大學碩士論文，2000。
[49]黃盈翔，「非奇異性邊界積分法對二維矩形流場之數值模擬」，國立台灣大學碩士論文，2004。
[50]廖健凱，「邊界元素法對二維翼型之流場分析」，國立台灣大學碩士論文，2011。
[51]施育宏，「利用最小方差法對二維翼型之跡流定位」，國立台灣大學碩士論文，2012。
[52]葉明學，「三維機翼尾端跡流之研究」，國立台灣大學碩士論文，2015。