拓扑流体力学及其新近发展

拓扑流体力学是理性力学的一个重要研究方向,在理论上具有重要价值,在实践中日益显出独特的作用.本文试对该方向进行综述性介绍,目的是吸引更多的国内科研工作者进入这一重要领域.本文的重点放在流体螺度(Helicity)的拓扑内涵方面.除了列出螺度与数学纽结场论的关系(即与互缠绕、自缠绕数以及作者近年来发展出来的流体纽结多项式拓扑不变量之间的关系),还介绍了国际上在流体纽结复杂系综的能量-结构复杂性关系方面的研究.最后,通过超流涡旋纽结/链环重联的具体实例,展示了这一领域当中典型的数值计算方法.我们希望通过这种理论推导与数值计算同时呈现的方式,使读者对这一国际前沿交叉领域的核心问题、研究方法以及科研中可能面对的技术困难获得一个整体的了解和把握.

Topological fluid mechanics and its new developments

Topological fluid mechanics is a crucial area of classical mechanics, with remarkable theoretical significance and wide applications in practice. In this paper, we expect to present an introductory review of the direction, for the purpose of attracting more domestic researchers of China to enter into this important area. Our emphasis is placed on the topological essence of fluid helicity. We will give the relationship between helicity and mathematical knotted field theory(i.e., that between helicity and the mutual-and self-linking numbers, as well as the fluid knot polynomial topological invariants constructed in terms of helicity recently discovered by the authors), and give an introduction to the international research on energy-structural complexity relationship for a fluid vortex knot ensemble. Moreover, typical numerical methods are also demonstrated through examples of reconnections occurring in superfluid vortex knots/links. Expectedly, a combination of the theoretical framework and numerical techniques may contribute to the reader a comprehensive understanding of the main target, research methodology and potential technical difficulties in practice in this interdisciplinary field of cutting-edge research world-wide.