管道和空腔中低雷诺数流问题的辛方法研究

There are many low Reynolds number flow problems in nature and engineering field. Analytical methods and semi-analytical methods are paid more attention because they can reveal the intrinsic laws of flow. However, the applications of these conventional analytical methods (semi-analytical methods) are restricted by some problems. Especially, most of these methods can be used to solve the linear low Reynolds number flow problems, but they are difficult to be applied to the nonlinear low Reynolds number flow problems. Therefore, it is necessary to develop a new method for low Reynolds number flow problems. In this project, we make efforts to study the flow in channels and cavities. We plan to firstly establish the Hamilton canonical equations and generalized Hamilton canonical equations for the linear and nonlinear low Reynolds number flow problems, respectively. In the symplectic system, the linear problem would be reduced to the eigenvalue and eigensolution problems, while the nonlinear problem would be investigated using a modal expansion method, so that the two kinds of problems can be both solved using the symplectic method. By means of these methods, we will carry on a comprehensive investigation into the low Reynolds number flow. The influence of the initial conditions and boundary conditions on the flow would be investigated in detail. This work may help us to discover some new phenomena and laws of the flow. Over all, we hope this project can develop a new semi-analytical method for solving low Reynolds number flow or other related fluid mechanics problems.

在自然界和工程技术领域存在着大量的低雷诺数流问题。解析和半解析法因能更真实地反映流动现象的本质规律而广被关注，但传统解析(半解析)法存在某些局限性，特别是大都集中在线性低雷诺数流问题上，难于求解更为一般的非线性低雷诺数流问题，因此迫切需要寻求一种新的求解方法。本项目以管道和空腔中低雷诺数流问题作为研究对象，针对线性和非线性低雷诺数流问题，分别建立哈密顿正则方程和广义哈密顿正则方程，在辛系统中将线性问题归结为辛本征值和辛本征解问题，将非线性问题采用模态展开方法，从而将低雷诺数流问题中的线性问题与非线性问题有机地结合起来，形成统一的辛求解方法。在此基础上，将对不同初、边值条件下的低雷诺数流问题开展深入研究，以期发现新的现象，揭示新的规律。通过本项目的实施有望形成一种新的半解析方法,为解决相关低雷诺数流和其他流体力学问题提供新途径。

自然界和工程技术领域存在着大量的低雷诺数流问题,准确高效地求解上述问题具有重要的工程和理论价值。解析和半解析法因能更真实地反映流动现象的本质规律而广被关注，但传统解析(半解析)法大都采用消元法，使微分方程的阶数提高，求解困难。此外，这些方法还难于处理边界条件，更难于推广到更为一般的非线性低雷诺数流问题。针对低雷诺数流问题的特点，本项目运用辛系统理论，建立了一种新的半解析求解方法。具体包括：(1)建立了平面Stokes流问题的辛方法，通过对偶变量的引入，不但解决了单一边界条件问题，还有效解决了混合边界条件处理方面的难题；(2)利用上述方法，对某些具体问题进行了计算，揭示了诸如弯管流和局部为自由应力表面时板驱动扇形空腔流等问题的规律和特点；(3)通过引入辛子体系，利用柱坐标系的特殊性，对变量进行二次分离，解决了三维柱坐标系下非零本征解求解困难的问题，建立了空间Stoke流问题的辛求解方法;(4)基于线性问题的正则方程，建立了非线性问题的广义哈密顿正则方程，再利用线性问题辛本征解空间的完备性，引入时间参数，采用模态展开法，给出了非线性问题解的形式及其展开系数的递推公式，建立了非线性问题的辛求解方法；(5) 通过拉普拉斯变换，消除方程中对时间的导数，对非定常线性低雷诺数流问题辛方法的建立进行了探讨。