摘 要：复变函数理论作为数学分析的重要分支，在流体动力学领域展现出独特的应用价值，为复杂流动问题的解析提供了有效工具。本研究以复变函数理论为基础，结合共形映射、奇异积分等方法，探索其在二维不可压缩流体流动建模与求解中的应用潜力。研究旨在通过构建复势函数模型，揭示流场中速度分布与压力场之间的内在联系，并针对典型边界条件下的流动问题提出高效解析方案。基于共形映射技术，本文成功将复杂几何区域转化为标准区域，从而显著简化了流动方程的求解过程。结果表明，复变函数方法能够精确描述绕流、涡旋等典型流动现象，并在多连通域问题中表现出优越性。本研究的主要创新点在于提出了一种适用于非规则边界条件的复势构造算法，为工程实际中的流动优化设计提供了理论支持，同时拓展了复变函数理论在现代流体力学中的应用范围。
关键词：复变函数理论；共形映射；二维不可压缩流动；复势函数；非规则边界条件
Exploration of the Application of Complex Functions in Fluid Dynamics
英文人名
Directive teacher:×××
Abstract：Complex variable theory, as an important branch of mathematical analysis, demonstrates unique application value in fluid dynamics, providing effective tools for solving complex flow problems. This study explores the potential of complex variable theory in modeling and solving two-dimensional incompressible fluid flows by integrating methods such as conformal mapping and singular integrals. The aim is to construct a complex potential function model to reveal the intrinsic relationship between velocity distribution and pressure fields in flow domains while proposing efficient analytical solutions for flow problems under typical boundary conditions. Leveraging conformal mapping techniques, this paper successfully transforms complex geometric regions into standard ones, significantly simplifying the solution process of flow equations. The results indicate that the complex variable method can accurately describe typical flow phenomena, such as flow around ob jects and vortices, and exhibits superiority in solving problems in multiply connected domains. A key innovation of this study is the development of a complex potential construction algorithm suitable for non-regular boundary conditions, offering theoretical support for flow optimization design in engineering applications and expanding the scope of complex variable theory in modern fluid mechanics.
Keywords: Theory Of Complex Functions；Conformal Mapping；Two-Dimensional Incompressible Flow；Complex Potential Function；Non-Regular Boundary Conditions
目 录
引言 1
一、复变函数的基础理论与流体动力学关联 1
(一)复变函数的基本概念 1
(二)流体动力学中的复势函数 2
(三)理论框架的构建与应用 2
二、复变函数在二维流动问题中的应用 3
(一)二维流动的复势表示 3
(二)拉普拉斯方程与保角映射 3
(三)典型流动问题的解析解 4
三、边界条件与复变函数的应用分析 4
(一)边界条件的数学描述 4
(二)保角映射在边界问题中的作用 5
(三)特殊边界条件下的流动模拟 5
四、复变函数在复杂流动中的拓展应用 6
(一)非理想流体中的复变方法 6
(二)多连通区域的流动分析 6
(三)数值方法与复变函数结合 7
结论 7
参考文献 9
致谢 9
